news_producer/vendor/markrogoyski/math-php/tests/LinearAlgebra/Matrix/Numeric/MatrixAxiomsTest.php

<?php

namespace MathPHP\Tests\LinearAlgebra\Matrix\Numeric;

use MathPHP\LinearAlgebra\MatrixFactory;
use MathPHP\LinearAlgebra\NumericSquareMatrix;
use MathPHP\LinearAlgebra\Vector;
use MathPHP\NumberTheory\Integer;
use MathPHP\Tests;

/**
 * Tests of Matrix axioms
 * These tests don't test specific functions,
 * but rather matrix axioms which in term make use of multiple functions.
 * If all the Matrix math is implemented properly, these tests should
 * all work out according to the axioms.
 *
 * Axioms tested:
 *  - Addition
 *    - r(A + B) = rA + rB
 *    - A + (−A) = 0
 *  - Multiplication
 *    - (AB)C = A(BC)
 *    - A(B + C) = AB + BC
 *    - r(AB) = (rA)B = A(rB)
 *  - Identity
 *    - AI = A = IA
 *    - I is involutory
 *  - Inverse
 *    - AA⁻¹ = I = A⁻¹A
 *    - (A⁻¹)⁻¹ = A
 *    - (AB)⁻¹ = B⁻¹A⁻¹
 *    - A is invertible, Aᵀ is inveritble
 *    - A is invertible, AAᵀ is inveritble
 *    - A is invertible, AᵀA is inveritble
 *  - Transpose
 *    - (Aᵀ)ᵀ = A
 *    - (A⁻¹)ᵀ = (Aᵀ)⁻¹
 *    - (rA)ᵀ = rAᵀ
 *    - (AB)ᵀ = BᵀAᵀ
 *    - (A + B)ᵀ = Aᵀ + Bᵀ
 *  - Trace
 *    - tr(A) = tr(Aᵀ)
 *    - tr(AB) = tr(BA)
 *  - Determinant
 *    - det(A) = det(Aᵀ)
 *    - det(AB) = det(A)det(B)
 *  - LU Decomposition (PA = LU)
 *    - PA = LU
 *    - A = P⁻¹LU
 *    - PPᵀ = I = PᵀP
 *    - (PA)⁻¹ = (LU)⁻¹ = U⁻¹L⁻¹
 *    - P⁻¹ = Pᵀ
 *  - QR Decomposition (A = QR)
 *    - A = QR
 *    - Q is orthogonal, R is upper triangular
 *    - QᵀQ = I
 *    - R = QᵀA
 *    - Qᵀ = Q⁻¹
 *  - Crout Decomposition (A = LU)
 *    - A = LU where L = LD
 *  - Cholesky Decomposition (A = LLᵀ)
 *     - A = LLᵀ
 *  - System of linear equations (Ax = b)
 *    - Ax - b = 0
 *    - x = A⁻¹b
 *    - LU: Ly = Pb and Ux = y
 *    - QR: x = R⁻¹Qᵀb
 *    - RREF of A augmented with B provides a solution to Ax = b
 *  - Symmetric matrix
 *    - A is square
 *    - A = Aᵀ
 *    - A⁻¹Aᵀ = I
 *    - A + B is symmetric
 *    - A - B is symmetric
 *    - kA is symmetric
 *    - AAᵀ is symmetric
 *    - AᵀA is symmetric
 *    - A is invertible symmetric, A⁻¹ is symmetric
 *  - Skew-symmetric matrix
 *    - The sum of two skew-symmetric matrices is skew-symmetric
 *    - Scalar multiple of a skew-symmetric matrix is skew-symmetric
 *    - The elements on the diagonal of a skew-symmetric matrix are zero, and therefore also its trace
 *    - A is skew-symmetric, then det(A) ≥ 0
 *    - A is a real skew-symmetric matrix, then I+A is invertible, where I is the identity matrix
 *  - Kronecker product
 *    - A ⊗ (B + C) = A ⊗ B + A ⊗ C
 *    - (A + B) ⊗ C = A ⊗ C + B ⊗ C
 *    - (A ⊗ B) ⊗ C = A ⊗ (B ⊗ C)
 *    - (A ⊗ B)(C ⊗ D) = AC ⊗ BD
 *    - (kA) ⊗ B = A ⊗ (kB) = k(A ⊗ B)
 *    - (A ⊗ B)⁻¹ = A⁻¹ ⊗ B⁻¹
 *    - (A ⊗ B)ᵀ = Aᵀ ⊗ Bᵀ
 *    - det(A ⊗ B) = det(A)ᵐ det(B)ⁿ
 *  - Kronecker sum
 *    - A⊕B = A⊗Ib + I⊗aB
 *  - Covariance matrix
 *    - S = Sᵀ
 *  - Positive definiteness
 *    - A is PD ⇔ -A is ND
 *    - A is PSD ⇔ -A is NSD
 *    - A is PD ⇒ A is PSD
 *    - A is ND ⇒ A is NSD
 *    - A is PD ⇔ A⁻¹ is PD
 *    - A is ND ⇔ A⁻¹ is ND
 *    - A is PD and r > 0 ⇒ rA is PD
 *    - A and B are PD ⇒ A + B is PD
 *    - A and B are PD ⇒ ABA is PD
 *    - A and B are PD ⇒ BAB is PD
 *  - Triangular
 *    - Zero matrix is lower triangular
 *    - Zero matrix is upper triangular
 *    - Lᵀ is upper triangular
 *    - Uᵀ is lower triangular
 *    - LL is lower triangular
 *    - UU is upper triangular
 *    - L + L is lower triangular
 *    - U + U is upper triangular
 *    - L⁻¹ is lower triangular (If L is invertible)
 *    - U⁻¹ is upper triangular (If U is invertible)
 *    - kL is lower triangular
 *    - kU is upper triangular
 *    - L is invertible iif diagonal is all non zero
 *    - U is invertible iif diagonal is all non zero
 *  - Diagonal
 *    - Zero matrix is diagonal
 *    - Dᵀ is diagonal
 *    - DD is diagonal
 *    - D + D is diagonal
 *    - D⁻¹ is diagonal (If D is invertible)
 *    - kD is lower triangular
 *    - D is invertible iif diagonal is all non zero
 *  - Reduced row echelon form
 *    - RREF is upper triangular
 *  - Exchange matrix
 *    - Jᵀ = J
 *    - J⁻¹ = J
 *    - tr(J) is 1 if n is odd, and 0 if n is even
 *  - Signature matrix
 *    - A is involutory
 *  - Hilbert matrix
 *    - H is symmetric
 *    - H is positive definite
 *  - Cholesky decomposition
 *    - A = LLᵀ
 *    - L is lower triangular
 *    - Lᵀ is upper triangular
 *  - Adjugate matrix
 *    - adj⟮A⟯ = Cᵀ
 *    - A adj⟮A⟯ = det⟮A⟯ I
 *    - A⁻¹ = (1/det⟮A⟯) adj⟮A⟯
 *    - adj⟮I⟯ = I
 *    - adj⟮AB⟯ = adj⟮B⟯adj⟮A⟯
 *    - adj⟮cA⟯ = cⁿ⁻¹ adj⟮A⟯
 *    - adj⟮B⟯adj⟮A⟯ = det⟮B⟯B⁻¹ det⟮A⟯A⁻¹ = det⟮AB⟯⟮AB⟯⁻¹
 *    - adj⟮Aᵀ⟯ = adj⟮A⟯ᵀ
 *    - Aadj⟮A⟯ = adj⟮A⟯A = det⟮A⟯I
 *  - Rank
 *    - rank(A) ≤ min(m, n)
 *    - Zero matrix has rank of 0
 *    - If A is square matrix, then it is invertible only if rank = n (full rank)
 *    - rank(AᵀA) = rank(AAᵀ) = rank(A) = rank(Aᵀ)
 *  - Bi/tridiagonal - Hessenberg
 *    - Lower bidiagonal matrix is upper Hessenberg
 *    - Upper bidiagonal matrix is lower Hessenberg
 *    - A matrix that is both upper Hessenberg and lower Hessenberg is a tridiagonal matrix
 *  - Orthogonal matrix
 *    - AAᵀ = I
 *    - AᵀA = I
 *    - A⁻¹ = Aᵀ
 *    - det(A) = 1
 *  - Householder matrix transformation
 *    - H is involutory
 *    - H has determinant that is -1
 *    - H has eigenvalues 1 and -1
 *  - Nilpotent matrix
 *    - tr(Aᵏ) = 0 for all k > 0
 *    - det(A) = 0
 *    - Cannot be invertible
 */
class MatrixAxiomsTest extends \PHPUnit\Framework\TestCase
{
    use Tests\LinearAlgebra\Fixture\MatrixDataProvider;

    /**
     * @test Axiom: r(A + B) = rA + rB
     * Order of scalar multiplication does not matter.
     *
     * @dataProvider dataProviderForScalarMultiplicationOrderAddition
     * @param        array $A
     * @param        array $B
     * @param        int $r
     * @throws       \Exception
     */
    public function testScalarMultiplicationOrderAddition(array $A, array $B, int $r)
    {
        // Given
        $A = MatrixFactory::create($A);
        $B = MatrixFactory::create($B);

        // When r(A + B)
        $A＋B = $A->add($B);
        $r⟮A＋B⟯ = $A＋B->scalarMultiply($r);

        // And rA + rB
        $rA     = $A->scalarMultiply($r);
        $rB     = $B->scalarMultiply($r);
        $rA＋rB = $rA->add($rB);

        // Then
        $this->assertEquals($r⟮A＋B⟯->getMatrix(), $rA＋rB->getMatrix());
    }

    /**
     * @return array
     */
    public function dataProviderForScalarMultiplicationOrderAddition(): array
    {
        return [
            [
                [
                    [1, 5],
                    [4, 3],
                ],
                [
                    [5, 6],
                    [2, 1],
                ], 5
            ],
            [
                [
                    [3, 8, 5],
                    [3, 6, 1],
                    [9, 5, 8],
                ],
                [
                    [5, 3, 8],
                    [6, 4, 5],
                    [1, 8, 9],
                ], 4
            ],
            [
                [
                    [-4, -2, 9],
                    [3, 14, -6],
                    [3, 9, 9],
                ],
                [
                    [8, 7, 8],
                    [-5, 4, 1],
                    [3, 5, 1],
                ], 7
            ],
            [
                [
                    [4, 7, 7, 8],
                    [3, 6, 4, 1],
                    [-3, 6, 8, -3],
                    [3, 2, 1, -54],
                ],
                [
                    [3, 2, 6, 7],
                    [4, 3, -6, 2],
                    [12, 14, 14, -6],
                    [4, 6, 4, -42],
                ], -8
            ],
        ];
    }

    /**
     * @test Axiom: A + (−A) = 0
     * Adding the negate of a matrix is a zero matrix.
     *
     * @dataProvider dataProviderForNegateAdditionZeroMatrix
     * @param        array $A
     * @param        array $Z
     * @throws       \Exception
     */
    public function testAddNegateIsZeroMatrix(array $A, array $Z)
    {
        // Given
        $A  = MatrixFactory::create($A);
        $Z  = MatrixFactory::create($Z);

        // When
        $−A = $A->negate();

        // Then
        $this->assertEquals($Z, $A->add($−A));
        $this->assertEquals($Z, $−A->add($A));
    }

    /**
     * @return array [A, Z]
     */
    public function dataProviderForNegateAdditionZeroMatrix(): array
    {
        return [
            [
                [
                    [0]
                ],
                [
                    [0]
                ]
            ],
            [
                [
                    [1]
                ],
                [
                    [0]
                ]
            ],
            [
                [
                    [1, 2, 3],
                    [4, 5, 6],
                    [7, 8, 9],
                ],
                [
                    [0, 0, 0],
                    [0, 0, 0],
                    [0, 0, 0],
                ],
            ],
            [
                [
                    [5, -4, 3, 2, -10],
                    [5, 5, 5, -4, -4],
                    [0, 0, -2, 4, 49],
                    [4, 3, 0, 0, -1],
                ],
                [
                    [0, 0, 0, 0, 0],
                    [0, 0, 0, 0, 0],
                    [0, 0, 0, 0, 0],
                    [0, 0, 0, 0, 0],
                ]
            ],
        ];
    }

    /**
     * @test Axiom: (AB)C = A(BC)
     * Matrix multiplication is associative
     *
     * @dataProvider dataProviderForMultiplicationIsAssociative
     * @param        array $A
     * @param        array $B
     * @param        array $C
     * @throws       \Exception
     */
    public function testMultiplicationIsAssociative(array $A, array $B, array $C)
    {
        // Given
        $A = MatrixFactory::create($A);
        $B = MatrixFactory::create($B);
        $C = MatrixFactory::create($C);

        // When
        $⟮AB⟯  = $A->multiply($B);
        $⟮AB⟯C = $⟮AB⟯->multiply($C);

        // And
        $⟮BC⟯  = $B->multiply($C);
        $A⟮BC⟯ = $A->multiply($⟮BC⟯);

        // Then
        $this->assertEquals($⟮AB⟯C->getMatrix(), $A⟮BC⟯->getMatrix());
    }

    /**
     * @return array
     */
    public function dataProviderForMultiplicationIsAssociative(): array
    {
        return [
            [
                [
                    [1, 5, 3],
                    [3, 6, 3],
                    [6, 7, 8],
                ],
                [
                    [6, 9, 9],
                    [3, 5, 1],
                    [3, 5, 12],
                ],
                [
                    [7, 4, 6],
                    [2, 3, 1],
                    [10, 12, 5],
                ],
            ],
            [
                [
                    [12, 21, 6],
                    [-3, 11, -6],
                    [3, 6, -3],
                ],
                [
                    [3, 7, 8],
                    [4, 4, 2],
                    [6, -4, 1],
                ],
                [
                    [1, -1, -5],
                    [6, 5, 19],
                    [3, 6, -2],
                ],
            ],
        ];
    }

    /**
     * @test Axiom: A(B + C) = AB + AC
     * Matrix multiplication is distributive
     *
     * @dataProvider dataProviderForMultiplicationIsDistributive
     * @param        array $A
     * @param        array $B
     * @param        array $C
     * @throws       \Exception
     */
    public function testMultiplicationIsDistributive(array $A, array $B, array $C)
    {
        // Given
        $A = MatrixFactory::create($A);
        $B = MatrixFactory::create($B);
        $C = MatrixFactory::create($C);

        // When A(B + C)
        $⟮B＋C⟯  = $B->add($C);
        $A⟮B＋C⟯ = $A->multiply($⟮B＋C⟯);

        // And AB + AC
        $AB     = $A->multiply($B);
        $AC     = $A->multiply($C);
        $AB＋AC = $AB->add($AC);

        // Then
        $this->assertEquals($A⟮B＋C⟯->getMatrix(), $AB＋AC->getMatrix());
    }

    /**
     * @return array
     */
    public function dataProviderForMultiplicationIsDistributive(): array
    {
        return [
            [
                [
                    [1, 2],
                    [0, -1],
                ],
                [
                    [0, -1],
                    [1, 1],
                ],
                [
                    [-2, 0],
                    [0, 1],
                ],
            ],
            [
                [
                    [1, 5, 3],
                    [3, 6, 3],
                    [6, 7, 8],
                ],
                [
                    [6, 9, 9],
                    [3, 5, 1],
                    [3, 5, 12],
                ],
                [
                    [7, 4, 6],
                    [2, 3, 1],
                    [10, 12, 5],
                ],
            ],
            [
                [
                    [12, 21, 6],
                    [-3, 11, -6],
                    [3, 6, -3],
                ],
                [
                    [3, 7, 8],
                    [4, 4, 2],
                    [6, -4, 1],
                ],
                [
                    [1, -1, -5],
                    [6, 5, 19],
                    [3, 6, -2],
                ],
            ],
        ];
    }

    /**
     * @test Axiom: r(AB) = (rA)B = A(rB)
     * Order of scalar multiplication does not matter.
     *
     * @dataProvider dataProviderForScalarMultiplicationOrder
     * @param        array $A
     * @param        array $B
     * @param        int $r
     * @throws       \Exception
     */
    public function testScalarMultiplcationOrder(array $A, array $B, int $r)
    {
        // Given
        $A = MatrixFactory::create($A);
        $B = MatrixFactory::create($B);

        // When r(AB)
        $AB = $A->multiply($B);
        $r⟮AB⟯ = $AB->scalarMultiply($r);

        // And (rA)B
        $rA = $A->scalarMultiply($r);
        $⟮rA⟯B = $rA->multiply($B);

        // And A(rB)
        $rB = $B->scalarMultiply($r);
        $A⟮rB⟯ = $A->multiply($rB);

        // Then
        $this->assertEquals($r⟮AB⟯->getMatrix(), $⟮rA⟯B->getMatrix());
        $this->assertEquals($⟮rA⟯B->getMatrix(), $A⟮rB⟯->getMatrix());
        $this->assertEquals($r⟮AB⟯->getMatrix(), $A⟮rB⟯->getMatrix());
    }

    /**
     * @return array
     */
    public function dataProviderForScalarMultiplicationOrder(): array
    {
        return [
            [
                [
                    [1, 5],
                    [4, 3],
                ],
                [
                    [5, 6],
                    [2, 1],
                ], 5
            ],
            [
                [
                    [3, 8, 5],
                    [3, 6, 1],
                    [9, 5, 8],
                ],
                [
                    [5, 3, 8],
                    [6, 4, 5],
                    [1, 8, 9],
                ], 4
            ],
            [
                [
                    [-4, -2, 9],
                    [3, 14, -6],
                    [3, 9, 9],
                ],
                [
                    [8, 7, 8],
                    [-5, 4, 1],
                    [3, 5, 1],
                ], 7
            ],
            [
                [
                    [4, 7, 7, 8],
                    [3, 6, 4, 1],
                    [-3, 6, 8, -3],
                    [3, 2, 1, -54],
                ],
                [
                    [3, 2, 6, 7],
                    [4, 3, -6, 2],
                    [12, 14, 14, -6],
                    [4, 6, 4, -42],
                ], -8
            ],
        ];
    }

    /**
     * @test Axiom: AI = A = IA
     * Matrix multiplied with the identity matrix is the original matrix.
     *
     * @dataProvider dataProviderForSquareMatrix
     * @param        array $A
     * @throws       \Exception
     */
    public function testMatrixTimesIdentityIsOriginalMatrix(array $A)
    {
        // Given
        $A  = MatrixFactory::create($A);
        $I  = MatrixFactory::identity($A->getN());

        // When
        $AI = $A->multiply($I);
        $IA = $I->multiply($A);

        // Then
        $this->assertEquals($A->getMatrix(), $AI->getMatrix());
        $this->assertEquals($A->getMatrix(), $IA->getMatrix());
    }

    /**
     * @test Axiom: I is involutory
     * Identity matrix is involutory
     * @throws \Exception
     */
    public function testIdentityMatrixIsInvolutory()
    {
        // Given
        foreach (\range(1, 20) as $n) {
            // When
            $A = MatrixFactory::identity($n);

            // Then
            $this->assertTrue($A->isInvolutory());
        }
    }

    /**
     * @test Axiom: AA⁻¹ = I = A⁻¹A
     * Matrix multiplied with its inverse is the identity matrix.
     *
     * @dataProvider dataProviderForInverse
     * @param        array $A
     * @throws       \Exception
     */
    public function testMatrixTimesInverseIsIdentity(array $A)
    {
        // Given
        $A = MatrixFactory::create($A);

        // When
        $A⁻¹  = $A->inverse();
        $AA⁻¹ = $A->multiply($A⁻¹);
        $A⁻¹A = $A⁻¹->multiply($A);
        $I    = MatrixFactory::identity($A->getN());

        // Then
        $this->assertEqualsWithDelta($I->getMatrix(), $AA⁻¹->getMatrix(), 0.00001);
        $this->assertEqualsWithDelta($I->getMatrix(), $A⁻¹A->getMatrix(), 0.00001);
        $this->assertEqualsWithDelta($AA⁻¹->getMatrix(), $A⁻¹A->getMatrix(), 0.00001);
    }

    /**
     * @test Axiom: (A⁻¹)⁻¹ = A
     * Inverse of inverse is the original matrix.
     *
     * @dataProvider dataProviderForSquareMatrixGreaterThanOneWithoutOddMatrices
     * @param        array $A
     * @throws       \Exception
     */
    public function testInverseOfInverseIsOriginalMatrix(array $A)
    {
        // Given
        $A = MatrixFactory::create($A);

        // When
        $⟮A⁻¹⟯⁻¹ = $A->inverse()->inverse();

        // Then
        $this->assertEqualsWithDelta($A->getMatrix(), $⟮A⁻¹⟯⁻¹->getMatrix(), 0.00001);
    }

    /**
     * @return array
     */
    public function dataProviderForInverse(): array
    {
        return [
            [
                [
                    [4, 7],
                    [2, 6],
                ],
                [
                    [0.6, -0.7],
                    [-0.2, 0.4],
                ],
            ],
            [
                [
                    [4, 3],
                    [3, 2],
                ],
                [
                    [-2, 3],
                    [3, -4],
                ],
            ],
            [
                [
                    [1, 2],
                    [3, 4],
                ],
                [
                    [-2, 1],
                    [3 / 2, -1 / 2],
                ],
            ],
            [
                [
                    [3, 3.5],
                    [3.2, 3.6],
                ],
                [
                    [-9, 8.75],
                    [8, -7.5],
                ],
            ],
            [
                [
                    [1, 2, 3],
                    [0, 4, 5],
                    [1, 0, 6],
                ],
                [
                    [12 / 11, -6 / 11, -1 / 11],
                    [5 / 22, 3 / 22, -5 / 22],
                    [-2 / 11, 1 / 11, 2 / 11],
                ],
            ],
            [
                [
                    [7, 2, 1],
                    [0, 3, -1],
                    [-3, 4, -2],
                ],
                [
                    [-2, 8, -5],
                    [3, -11, 7],
                    [9, -34, 21],
                ],
            ],
            [
                [
                    [3, 6, 6, 8],
                    [4, 5, 3, 2],
                    [2, 2, 2, 3],
                    [6, 8, 4, 2],
                ],
                [
                    [-0.333, 0.667, 0.667, -0.333],
                    [0.167, -2.333, 0.167, 1.417],
                    [0.167, 4.667, -1.833, -2.583],
                    [0.000, -2.000, 1.000, 1.000],
                ],
            ],
            [
                [
                    [4, 23, 6, 4, 7],
                    [3, 64, 23, 52, 2],
                    [65, 45, 3, 23, 1],
                    [2, 3, 4, 3, 9],
                    [53, 99, 54, 32, 105],
                ],
                [
                    [-0.142, 0.006, 0.003, -0.338, 0.038],
                    [0.172, -0.012, 0.010, 0.275, -0.035],
                    [-0.856, 0.082, -0.089, -2.344, 0.257],
                    [0.164, -0.001, 0.026, 0.683, -0.070],
                    [0.300, -0.033, 0.027, 0.909, -0.088],
                ],
            ],
        ];
    }

    /**
     * (AB)⁻¹ = B⁻¹A⁻¹
     * The inverse of a product is the reverse product of the inverses.
     *
     * @dataProvider dataProviderForInverseProductIsReverseProductOfInverses
     * @param        array $A
     * @param        array $B
     * @throws       \Exception
     */
    public function testInverseProductIsReverseProductOfInverses(array $A, array $B)
    {
        // Given
        $A = MatrixFactory::create($A);
        $B = MatrixFactory::create($B);

        // When
        $⟮AB⟯⁻¹ = $A->multiply($B)->inverse();

        // And
        $B⁻¹ = $B->inverse();
        $A⁻¹ = $A->inverse();
        $B⁻¹A⁻¹ = $B⁻¹->multiply($A⁻¹);

        // Then
        $this->assertEqualsWithDelta($⟮AB⟯⁻¹->getMatrix(), $B⁻¹A⁻¹->getMatrix(), 0.00001);
    }

    /**
     * @test Axiom: A is invertible, Aᵀ is inveritble
     * If A is an invertible matrix, then the transpose is also inveritble
     * @dataProvider dataProviderForInverse
     * @param        array $A
     * @throws       \Exception
     */
    public function testIfMatrixIsInvertibleThenTransposeIsInvertible(array $A)
    {
        // Given
        $A = MatrixFactory::create($A);

        // When
        $Aᵀ = $A->transpose();

        // Then
        if ($A->isInvertible()) {
            $this->assertTrue($Aᵀ->isInvertible());
        } else {
            $this->assertFalse($Aᵀ->isInvertible());
        }
    }

    /**
     * @test Axiom: A is invertible, AAᵀ is inveritble
     * If A is an invertible matrix, then the product of A and tranpose of A is also inveritble
     * @dataProvider dataProviderForInverse
     * @param        array $A
     * @throws       \Exception
     */
    public function testIfMatrixIsInvertibleThenProductOfMatrixAndTransposeIsInvertible(array $A)
    {
        // Given
        $A = MatrixFactory::create($A);

        // When
        $Aᵀ  = $A->transpose();
        $AAᵀ = $A->multiply($Aᵀ);

        // then
        if ($A->isInvertible()) {
            $this->assertTrue($AAᵀ->isInvertible());
        } else {
            $this->assertFalse($AAᵀ->isInvertible());
        }
    }

    /**
     * @test Axiom: A is invertible, AᵀA is inveritble
     * If A is an invertible matrix, then the product of transpose and A is also inveritble
     * @dataProvider dataProviderForInverse
     * @param        array $A
     * @throws       \Exception
     */
    public function testIfMatrixIsInvertibleThenProductOfTransposeAndMatrixIsInvertible(array $A)
    {
        // Given
        $A = MatrixFactory::create($A);

        // When
        $Aᵀ  = $A->transpose();
        $AᵀA = $Aᵀ->multiply($A);

        // Then
        if ($A->isInvertible()) {
            $this->assertTrue($AᵀA->isInvertible());
        } else {
            $this->assertFalse($AᵀA->isInvertible());
        }
    }

    /**
     * @return array
     */
    public function dataProviderForInverseProductIsReverseProductOfInverses(): array
    {
        return [
            [
                [
                    [1, 5],
                    [4, 3],
                ],
                [
                    [5, 6],
                    [2, 1],
                ],
            ],
            [
                [
                    [3, 8, 5],
                    [3, 6, 1],
                    [9, 5, 8],
                ],
                [
                    [5, 3, 8],
                    [6, 4, 5],
                    [1, 8, 9],
                ],
            ],
            [
                [
                    [-4, -2, 9],
                    [3, 14, -6],
                    [3, 9, 9],
                ],
                [
                    [8, 7, 8],
                    [-5, 4, 1],
                    [3, 5, 1],
                ],
            ],
            [
                [
                    [4, 7, 7, 8],
                    [3, 6, 4, 1],
                    [-3, 6, 8, -3],
                    [3, 2, 1, -54],
                ],
                [
                    [3, 2, 6, 7],
                    [4, 3, -6, 2],
                    [12, 14, 14, -6],
                    [4, 6, 4, -42],
                ],
            ],
        ];
    }

    /**
     * @test Axiom: (Aᵀ)ᵀ = A
     * The transpose of the transpose is the original matrix.
     *
     * @dataProvider dataProviderForSquareMatrix
     * @param        array $A
     * @throws       \Exception
     */
    public function testTransposeOfTransposeIsOriginalMatrix(array $A)
    {
        // Given
        $A = MatrixFactory::create($A);

        // When
        $⟮A⁻ᵀ⟯ᵀ = $A->transpose()->transpose();

        // Then
        $this->assertEquals($⟮A⁻ᵀ⟯ᵀ->getMatrix(), $A->getMatrix());
    }

    /**
     * @test Axiom: (A⁻¹)ᵀ = (Aᵀ)⁻¹
     * The transpose of the inverse is the inverse of the transpose.
     *
     * @dataProvider dataProviderForSquareMatrixGreaterThanOneWithoutOddMatrices
     * @param        array $A
     * @throws       \Exception
     */
    public function testTransposeOfInverseIsInverseOfTranspose(array $A)
    {
        // Given
        $A = MatrixFactory::create($A);

        // When
        $⟮A⁻¹⟯ᵀ = $A->inverse()->transpose();
        $⟮Aᵀ⟯⁻¹ = $A->transpose()->inverse();

        // Then
        $this->assertEqualsWithDelta($⟮A⁻¹⟯ᵀ->getMatrix(), $⟮Aᵀ⟯⁻¹->getMatrix(), 0.00001);
    }

    /**
     * @test Axiom: (rA)ᵀ = rAᵀ
     * Scalar multiplication order does not matter for transpose
     *
     * @dataProvider dataProviderForSquareMatrix
     * @param        array $A
     * @throws       \Exception
     */
    public function testScalarMultiplicationOfTransposeOrder(array $A)
    {
        // Given
        $A = MatrixFactory::create($A);
        $r = 4;

        // When
        $⟮rA⟯ᵀ = $A->scalarMultiply($r)->transpose();
        $rAᵀ  = $A->transpose()->scalarMultiply($r);

        // Then
        $this->assertEquals($⟮rA⟯ᵀ->getMatrix(), $rAᵀ->getMatrix());
    }

    /**
     * @test Axiom: (AB)ᵀ = BᵀAᵀ
     * Transpose of a product of matrices equals the product of their transposes in reverse order.
     *
     * @dataProvider dataProviderForTwoSquareMatrices
     * @param        array $A
     * @param        array $B
     * @throws       \Exception
     */
    public function testTransposeProductIsProductOfTranposesInReverseOrder(array $A, array $B)
    {
        // Given
        $A = MatrixFactory::create($A);
        $B = MatrixFactory::create($B);

        // When (AB)ᵀ
        $⟮AB⟯ᵀ = $A->multiply($B)->transpose();

        // And BᵀAᵀ
        $Bᵀ   = $B->transpose();
        $Aᵀ   = $A->transpose();
        $BᵀAᵀ = $Bᵀ->multiply($Aᵀ);

        // Then
        $this->assertEquals($⟮AB⟯ᵀ->getMatrix(), $BᵀAᵀ->getMatrix());
    }

    /**
     * @test Axiom: (A + B)ᵀ = Aᵀ + Bᵀ
     * Transpose of sum is the same as sum of transposes
     *
     * @dataProvider dataProviderForTwoSquareMatrices
     * @param        array $A
     * @param        array $B
     * @throws       \Exception
     */
    public function testTransposeSumIsSameAsSumOfTransposes(array $A, array $B)
    {
        // Given
        $A = MatrixFactory::create($A);
        $B = MatrixFactory::create($B);

        // When (A + B)ᵀ
        $⟮A＋B⟯ᵀ = $A->add($B)->transpose();

        // And Aᵀ + Bᵀ
        $Aᵀ     = $A->transpose();
        $Bᵀ     = $B->transpose();
        $Aᵀ＋Bᵀ = $Aᵀ->add($Bᵀ);

        // Then
        $this->assertEquals($⟮A＋B⟯ᵀ->getMatrix(), $Aᵀ＋Bᵀ->getMatrix());
    }

    /**
     * @test Axiom: tr(A) = tr(Aᵀ)
     * Trace is the same as the trace of the transpose
     *
     * @dataProvider dataProviderForSquareMatrix
     * @param        array $A
     * @throws       \Exception
     */
    public function testTraceIsSameAsTraceOfTranspose(array $A)
    {
        // Given
        $A  = MatrixFactory::create($A);

        // When
        $Aᵀ = $A->transpose();
        $tr⟮A⟯  = $A->trace();
        $tr⟮Aᵀ⟯ = $Aᵀ->trace();

        // Then
        $this->assertEquals($tr⟮A⟯, $tr⟮Aᵀ⟯);
    }

    /**
     * @test Axiom: tr(AB) = tr(BA)
     * Trace of product does not matter the order they were multiplied
     *
     * @dataProvider dataProviderForTwoSquareMatrices
     * @param        array $A
     * @param        array $B
     * @throws       \Exception
     */
    public function testTraceOfProductIsSameRegardlessOfOrderMultiplied(array $A, array $B)
    {
        // Given
        $A  = MatrixFactory::create($A);
        $B  = MatrixFactory::create($B);

        // When
        $tr⟮AB⟯ = $A->multiply($B)->trace();
        $tr⟮BA⟯ = $B->multiply($A)->trace();

        // Then
        $this->assertEquals($tr⟮AB⟯, $tr⟮BA⟯);
    }

    /**
     * @test Axiom: det(A) = det(Aᵀ)
     * Determinant of matrix is the same as determinant of transpose.
     *
     * @dataProvider dataProviderForSquareMatrix
     * @param        array $A
     * @throws       \Exception
     */
    public function testDeterminantSameAsDeterminantOfTranspose(array $A)
    {
        // Given
        $A = MatrixFactory::create($A);

        // When
        $det⟮A⟯  = $A->det();
        $det⟮Aᵀ⟯ = $A->transpose()->det();

        // Then
        $this->assertEqualsWithDelta($det⟮A⟯, $det⟮Aᵀ⟯, 0.00001);
    }

    /**
     * @test Axiom: det(AB) = det(A)det(B)
     * Determinant of product of matrices is the same as the product of determinants.
     *
     * @dataProvider dataProviderForTwoSquareMatrices
     * @param        array $A
     * @param        array $B
     * @throws       \Exception
     */
    public function testDeterminantProductSameAsProductOfDeterminants(array $A, array $B)
    {
        // Given
        $A = MatrixFactory::create($A);
        $B = MatrixFactory::create($B);

        // When det(AB)
        $det⟮AB⟯  = $A->multiply($B)->det();

        // And det(A)det(B)
        $det⟮A⟯ = $A->det();
        $det⟮B⟯ = $B->det();
        $det⟮A⟯det⟮B⟯ = $det⟮A⟯ * $det⟮B⟯;

        // Then
        $this->assertEqualsWithDelta($det⟮AB⟯, $det⟮A⟯det⟮B⟯, 0.000001);
    }

    /**
     * @test Axiom: PA = LU
     * Basic LU decomposition property that permutation matrix times the matrix is the product of the lower and upper decomposition matrices.
     * @dataProvider dataProviderForSquareMatrixGreaterThanOneWithoutOddMatrices
     * @dataProvider dataProviderForSymmetricMatrix
     * @param        array $A
     * @throws       \Exception
     */
    public function testLUDecompositionPAEqualsLU(array $A)
    {
        // Given
        $A = MatrixFactory::create($A);

        // When
        $LUP = $A->luDecomposition();
        $L   = $LUP->L;
        $U   = $LUP->U;
        $P   = $LUP->P;

        // Then PA = LU;
        $PA = $P->multiply($A);
        $LU = $L->multiply($U);
        $this->assertEqualsWithDelta($PA->getMatrix(), $LU->getMatrix(), 0.00001);
    }

    /**
     * @test Axiom: A = P⁻¹LU
     * @dataProvider dataProviderForSquareMatrixGreaterThanOneWithoutOddMatrices
     * @dataProvider dataProviderForSymmetricMatrix
     * @param        array $A
     * @throws       \Exception
     */
    public function testLUDecompositionAEqualsPInverseLU(array $A)
    {
        // Given
        $A = MatrixFactory::create($A);

        // When
        $LUP = $A->luDecomposition();
        $L   = $LUP->L;
        $U   = $LUP->U;
        $P   = $LUP->P;

        // Then A = P⁻¹LU
        $P⁻¹LU = $P->inverse()->multiply($L)->multiply($U);
        $this->assertEqualsWithDelta($A->getMatrix(), $P⁻¹LU->getMatrix(), 0.00001);
    }

    /**
     * @test Axiom: PPᵀ = I = PᵀP
     * Permutation matrix of the LU decomposition times the transpose of the permutation matrix is the identity matrix.
     * @dataProvider dataProviderForSquareMatrix
     * @dataProvider dataProviderForSymmetricMatrix
     * @param        array $A
     * @throws       \Exception
     */
    public function testLUDecompositionPPTransposeEqualsIdentity(array $A)
    {
        // Given
        $A = MatrixFactory::create($A);

        // When
        $LUP = $A->luDecomposition();

        $P  = $LUP->P;
        $Pᵀ = $P->transpose();

        $PPᵀ = $P->multiply($Pᵀ);
        $PᵀP = $Pᵀ->multiply($P);

        $I = MatrixFactory::identity($A->getM());

        // Then PPᵀ = I = PᵀP
        $this->assertEquals($PPᵀ->getMatrix(), $I->getMatrix());
        $this->assertEquals($I->getMatrix(), $PᵀP->getMatrix());
        $this->assertEquals($PPᵀ->getMatrix(), $PᵀP->getMatrix());
    }

    /**
     * @test Axiom: (PA)⁻¹ = (LU)⁻¹ = U⁻¹L⁻¹
     * Inverse of the LU decomposition equation is the inverse of the other side.
     * @dataProvider dataProviderForSquareMatrixGreaterThanOneWithoutOddMatrices
     * @dataProvider dataProviderForSymmetricMatrix
     * @param        array $A
     * @throws       \Exception
     */
    public function testInverseWithLUDecompositionInverse(array $A)
    {
        // Given
        $A = MatrixFactory::create($A);

        // When
        $LUP = $A->luDecomposition();
        $L   = $LUP->L;
        $U   = $LUP->U;
        $P   = $LUP->P;

        $⟮PA⟯⁻¹ = $P->multiply($A)->inverse();
        $⟮LU⟯⁻¹ = $L->multiply($U)->inverse();

        $U⁻¹    = $U->inverse();
        $L⁻¹    = $L->inverse();
        $U⁻¹L⁻¹ = $U⁻¹->multiply($L⁻¹);

        // Then (PA)⁻¹ = (LU)⁻¹ = U⁻¹L⁻¹
        $this->assertEqualsWithDelta($⟮PA⟯⁻¹->getMatrix(), $⟮LU⟯⁻¹->getMatrix(), 0.00001);
        $this->assertEqualsWithDelta($⟮LU⟯⁻¹->getMatrix(), $U⁻¹L⁻¹->getMatrix(), 0.00001);
        $this->assertEqualsWithDelta($⟮PA⟯⁻¹->getMatrix(), $U⁻¹L⁻¹->getMatrix(), 0.00001);
    }

    /**
     * @test Axiom: A = QR
     * Basic QR decomposition property that A = QR
     * @dataProvider dataProviderForSquareMatrix
     * @dataProvider dataProviderForNotSquareMatrix
     * @dataProvider dataProviderForSymmetricMatrix
     * @dataProvider dataProviderForNotSymmetricMatrix
     * @param        array $A
     * @throws       \Exception
     */
    public function testQRDecompositionAEqualsQR(array $A)
    {
        // Given
        $A = MatrixFactory::create($A);

        // When
        $qr = $A->qrDecomposition();
        $Q  = $qr->Q;
        $R  = $qr->R;

        // Then A = QR
        $this->assertEqualsWithDelta($A->getMatrix(), $Q->multiply($R)->getMatrix(), 0.00001);
    }

    /**
     * @test Axiom: QR.Q is orthogonal and QR.R is upper triangular
     * QR decomposition properties Q is orthogonal and R is upper triangular
     * @dataProvider dataProviderForSquareMatrix
     * @dataProvider dataProviderForSymmetricMatrix
     * @param        array $A
     * @throws       \Exception
     */
    public function testQRDecompositionQOrthogonalRUpperTriangular(array $A)
    {
        // Given
        $A = MatrixFactory::create($A);

        // When
        $qr  = $A->qrDecomposition();

        // Then Q is orthogonal and R is upper triangular
        $this->assertTrue($qr->Q->isOrthogonal());
        $this->assertTrue($qr->R->isUpperTriangular());
    }

    /**
     * @test         Axiom QᵀQ = I
     *               QR decomposition property orthonormal matrix Q has the property QᵀQ = I
     * @dataProvider dataProviderForSquareMatrix
     * @dataProvider dataProviderForNotSquareMatrix
     * @dataProvider dataProviderForSymmetricMatrix
     * @dataProvider dataProviderForNotSymmetricMatrix
     * @param        array $A
     * @throws       \Exception
     */
    public function testQrDecompositionOrthonormalMatrixQPropertyQTransposeQIsIdentity(array $A)
    {
        // Given
        $A = MatrixFactory::create($A);
        $I = MatrixFactory::identity(\min($A->getM(), $A->getN()));

        // And
        $qr = $A->qrDecomposition();

        // When
        $QᵀQ = $qr->Q->transpose()->multiply($qr->Q);

        // Then QᵀQ = I
        $this->assertEqualsWithDelta($I->getMatrix(), $QᵀQ->getMatrix(), 0.000001);
    }

    /**
     * @test         Axiom R = QᵀA
     *               QR decomposition property R = QᵀA
     * @dataProvider dataProviderForSquareMatrix
     * @dataProvider dataProviderForNotSquareMatrix
     * @dataProvider dataProviderForSymmetricMatrix
     * @dataProvider dataProviderForNotSymmetricMatrix
     * @param        array $A
     * @throws       \Exception
     */
    public function testQrDecompositionPropertyREqualsQTransposeA(array $A)
    {
        // Given
        $A = MatrixFactory::create($A);

        // And
        $qrDecomposition = $A->qrDecomposition();

        // When
        $QᵀA = $qrDecomposition->Q->transpose()->multiply($A);

        // Then R = QᵀA
        $this->assertEqualsWithDelta($qrDecomposition->R->getMatrix(), $QᵀA->getMatrix(), 0.00001);
    }

    /**
     * @test         Axiom Qᵀ = Q⁻¹
     *               QR decomposition property Qᵀ = Q⁻¹
     * @dataProvider dataProviderForSquareMatrix
     * @dataProvider dataProviderForSymmetricMatrix
     * @param        array $A
     * @throws       \Exception
     */
    public function testQrDecompositionPropertyQTransposeEqualsQInverse(array $A)
    {
        // Given
        $A = MatrixFactory::create($A);

        // And
        $Q = $A->qrDecomposition()->Q;

        // When
        $Qᵀ  = $Q->transpose();
        $Q⁻¹ = $Q->inverse();

        // Then Qᵀ = Q⁻¹
        $this->assertEqualsWithDelta($Qᵀ->getMatrix(), $Q⁻¹->getMatrix(), 0.00001);
    }

    /**
     * @test Axiom: A = LU where L = LD
     * Basic Crout decomposition property that A = (LD)U
     * @dataProvider dataProviderForSquareMatrixGreaterThanOneWithoutOddMatrices
     * @dataProvider dataProviderForSymmetricMatrix
     * @param        array $A
     * @throws       \Exception
     */
    public function testCroutDecompositionAEqualsLDU(array $A)
    {
        // Given
        $A = MatrixFactory::create($A);

        // When
        $crout = $A->croutDecomposition();
        $L     = $crout->L;
        $U     = $crout->U;

        // Then A = LU
        $this->assertEqualsWithDelta($A->getMatrix(), $L->multiply($U)->getMatrix(), 0.00001);
    }

    /**
     * @test Axiom: A = LLᵀ
     * Basic Cholesky decomposition property that A = LLᵀ
     * @dataProvider dataProviderForPositiveDefiniteMatrix
     * @param        array $A
     * @throws       \Exception
     */
    public function testCholeskyDecompositionAEqualsLLᵀ(array $A)
    {
        // Given
        $A = MatrixFactory::create($A);

        // When
        $cholesky = $A->choleskyDecomposition();
        $L        = $cholesky->L;
        $Lᵀ       = $cholesky->LT;

        // Then A = LLᵀ
        $this->assertEqualsWithDelta($A->getMatrix(), $L->multiply($Lᵀ)->getMatrix(), 0.00001);
    }

    /**
     * @test Axiom: P⁻¹ = Pᵀ
     * Inverse of the permutation matrix equals the transpose of the permutation matrix
     *
     * @dataProvider dataProviderForSquareMatrix
     * @param        array $A
     * @throws       \Exception
     */
    public function testPInverseEqualsPTranspose(array $A)
    {
        // Given
        $A = MatrixFactory::create($A);

        // When
        $LUP = $A->luDecomposition();
        $P   = $LUP->P;
        $P⁻¹ = $P->inverse();
        $Pᵀ  = $P->transpose();

        // Then
        $this->assertEquals($P⁻¹, $Pᵀ);
    }

    /**
     * @test Axiom: Ax - b = 0
     * Matrix multiplied with unknown x vector subtract solution b is 0
     *
     * @dataProvider dataProviderForSolve
     * @param        array $A
     * @param        array $b
     * @param        array $x
     * @throws       \Exception
     */
    public function testSolveEquationForZero(array $A, array $b, array $x)
    {
        // Given
        $A = MatrixFactory::create($A);
        $x = new Vector($x);
        $b = (new Vector($b))->asColumnMatrix();

        // And zeros
        $z = (new Vector(\array_fill(0, count($x), 0)))->asColumnMatrix();

        // When Ax - b
        $R = $A->multiply($x)->subtract($b);

        // Then
        $this->assertEqualsWithDelta($z, $R, 0.01);
    }

    /**
     * @test Axiom: x = A⁻¹b
     * Inverse of A multiplied with b is a solution to x
     *
     * @dataProvider dataProviderForSolve
     * @param        array $A
     * @param        array $b
     * @param        array $x
     * @throws       \Exception
     */
    public function testSolveInverseBEqualsX(array $A, array $b, array $x)
    {
        // Given
        $A   = MatrixFactory::create($A);
        $A⁻¹ = $A->inverse();
        $x = (new Vector($x))->asColumnMatrix();
        $b = new Vector($b);

        // When A⁻¹b
        $A⁻¹b = $A⁻¹->multiply($b);

        // Then
        $this->assertEqualsWithDelta($x, $A⁻¹b, 0.001);
    }

    /**
     * @test Axiom: LU: Ly = Pb and Ux = y
     * LU decomposition provides a solution to Ax = b
     *
     * @dataProvider dataProviderForSolve
     * @param        array $A
     * @param        array $b
     * @param        array $expected
     * @throws       \Exception
     */
    public function testSolveLuDecomposition(array $A, array $b, array $expected)
    {
        // Given
        $A        = MatrixFactory::create($A);
        $expected = new Vector($expected);

        // And
        $LU = $A->luDecomposition();

        // When Ly = Pb and Ux = y
        $x = $LU->solve($b);

        // Then
        $this->assertEqualsWithDelta($expected, $x, 0.001);
    }

    /**
     * @test Axiom: QR: x = R⁻¹Qᵀb
     * QR decomposition provides a solution to Ax = b
     *
     * @dataProvider dataProviderForSolve
     * @param        array $A
     * @param        array $b
     * @param        array $expected
     * @throws       \Exception
     */
    public function testSolveQrDecomposition(array $A, array $b, array $expected)
    {
        // Given
        $A        = MatrixFactory::create($A);
        $expected = new Vector($expected);

        // And
        $QR = $A->qrDecomposition();

        // When x = R⁻¹Qᵀb
        $x = $QR->solve($b);

        // Then
        $this->assertEqualsWithDelta($expected, $x, 0.001);
    }

    /**
     * @test Axiom: RREF of A augmented with B provides a solution to Ax = b
     *
     * @dataProvider dataProviderForSolve
     * @param        array $A
     * @param        array $b
     * @param        array $expected
     * @throws       \Exception
     */
    public function testSolveRref(array $A, array $b, array $expected)
    {
        // Given
        $A        = MatrixFactory::create($A);
        $b        = new Vector($b);
        $expected = new Vector($expected);

        // And
        $QR = $A->qrDecomposition();

        // When RREF
        $Ab   = $A->augment($b->asColumnMatrix());
        $rref = $Ab->rref();
        $x    = new Vector(\array_column($rref->getMatrix(), $rref->getN() - 1));

        // Then
        $this->assertEqualsWithDelta($expected, $x, 0.001);
    }

    /**
     * @test Axiom: Symmetric matrix is square
     * @dataProvider dataProviderForSymmetricMatrix
     * @param        array $A
     * @throws       \Exception
     */
    public function testSymmetricMatrixIsSquare(array $A)
    {
        // Given
        $A = MatrixFactory::create($A);

        // When
        $isSquare = $A->isSquare();

        // Then
        $this->assertTrue($isSquare);
    }

    /**
     * @test Axiom: A = Aᵀ
     * Symmetric matrix is the same as its transpose
     * @dataProvider dataProviderForSymmetricMatrix
     * @param        array $A
     * @throws       \Exception
     */
    public function testSymmetricEqualsTranspose(array $A)
    {
        // Given
        $A = MatrixFactory::create($A);

        // When
        $Aᵀ = $A->transpose();

        // Then
        $this->assertEquals($A->getMatrix(), $Aᵀ->getMatrix());
    }

    /**
     * @test Axiom: A⁻¹Aᵀ = I
     * Symmetric matrix inverse times transpose equals identity matrix
     * @dataProvider dataProviderForSymmetricMatrix
     * @param        array $A
     * @throws       \Exception
     */
    public function testSymmetricInverseTransposeEqualsIdentity(array $A)
    {
        // Given
        $A = MatrixFactory::create($A);

        // When
        $A⁻¹ = $A->inverse();
        $Aᵀ  = $A->transpose();

        // And
        $A⁻¹Aᵀ = $A⁻¹->multiply($Aᵀ);
        $I     = MatrixFactory::identity($A->getM());

        // Then
        $this->assertEqualsWithDelta($I, $A⁻¹Aᵀ, 0.00001);
        $this->assertEqualsWithDelta($I->getMatrix(), $A⁻¹Aᵀ->getMatrix(), 0.00001);
    }

    /**
     * @test Axiom: A + B is symmetric
     * If A and B are symmetric matrices with the sme size, then A + B is symmetric
     * @dataProvider dataProviderForSymmetricMatrix
     * @param        array $M
     * @throws       \Exception
     */
    public function testSymmetricMatricesSumIsSymmetric(array $M)
    {
        // Given
        $A  = MatrixFactory::create($M);
        $B  = MatrixFactory::create($M);

        // When
        $A＋B = $A->add($B);

        // Then
        $this->assertTrue($A->isSymmetric());
        $this->assertTrue($B->isSymmetric());
        $this->assertTrue($A＋B->isSymmetric());
    }

    /**
     * @test Axiom: A - B is symmetric
     * If A and B are symmetric matrices with the sme size, then A - B is symmetric
     * @dataProvider dataProviderForSymmetricMatrix
     * @param        array $M
     * @throws       \Exception
     */
    public function testSymmetricMatricesDifferenceIsSymmetric(array $M)
    {
        // Given
        $A = MatrixFactory::create($M);
        $B = MatrixFactory::create($M);

        // When
        $A−B = $A->subtract($B);

        // Then
        $this->assertTrue($A->isSymmetric());
        $this->assertTrue($B->isSymmetric());
        $this->assertTrue($A−B->isSymmetric());
    }

    /**
     * @test Axiom: kA is symmetric
     * If A is a symmetric matrix, kA is symmetric
     * @dataProvider dataProviderForSymmetricMatrix
     * @param        array $M
     * @throws       \Exception
     */
    public function testSymmetricMatricesTimesScalarIsSymmetric(array $M)
    {
        // Given
        $A   = MatrixFactory::create($M);
        $this->assertTrue($A->isSymmetric());

        foreach (\range(1, 10) as $k) {
            // When
            $kA = $A->scalarMultiply($k);

            // Then
            $this->assertTrue($kA->isSymmetric());
        }
    }

    /**
     * @test Axiom: AAᵀ is symmetric
     * If A is a symmetric matrix, AAᵀ is symmetric
     * @dataProvider dataProviderForSymmetricMatrix
     * @param        array $M
     * @throws       \Exception
     */
    public function testSymmetricMatrixTimesTransposeIsSymmetric(array $M)
    {
        // Given
        $A = MatrixFactory::create($M);

        // When
        $Aᵀ  = $A->transpose();
        $AAᵀ = $A->multiply($Aᵀ);

        // Then
        $this->assertTrue($A->isSymmetric());
        $this->assertTrue($AAᵀ->isSymmetric());
    }

    /**
     * @test Axiom: AᵀA is symmetric
     * If A is a symmetric matrix, AᵀA is symmetric
     * @dataProvider dataProviderForSymmetricMatrix
     * @param        array $M
     * @throws       \Exception
     */
    public function testTransposeTimesSymmetricMatrixIsSymmetric(array $M)
    {
        // Given
        $A = MatrixFactory::create($M);

        // When
        $Aᵀ  = $A->transpose();
        $AᵀA = $Aᵀ->multiply($A);

        // Then
        $this->assertTrue($A->isSymmetric());
        $this->assertTrue($AᵀA->isSymmetric());
    }

    /**
     * @test Axiom: A is invertible symmetric, A⁻¹ is symmetric
     * If A is an invertible symmetric matrix, the inverse of A is also symmetric
     * @dataProvider dataProviderForSymmetricMatrix
     * @param        array $M
     * @throws       \Exception
     */
    public function testMatrixIsInvertibleSymmetricThenInverseIsSymmetric(array $M)
    {
        // Given
        $A   = MatrixFactory::create($M);

        if ($A->isInvertible() && $A->isSymmetric()) {
            // When
            $A⁻¹ = $A->inverse();
            $A⁻¹ = $A⁻¹->map(
                function ($x) {
                    return \round($x, 5); // Floating point adjustment
                }
            );

            // Theb
            $this->assertTrue($A⁻¹->isSymmetric());
        }
    }

    /**
     * @test Axiom: A is skew-symmetric, det(A) ≥ 0
     *               If A is a skew-symmetric matrix, the determinant is greater than zero.
     * @dataProvider dataProviderForSkewSymmetricMatrix
     * @param        array $A
     * @throws       \Exception
     */
    public function testMatrixIsSkewSymmetricDeterminantGreaterThanZero(array $A)
    {
        // Given
        $A = MatrixFactory::create($A);

        // Then
        $this->assertTrue($A->isSkewSymmetric());
        $this->assertGreaterThanOrEqual(0, $A->det());
    }

    /**
     * @test Axiom: The sum of two skew-symmetric matrices is skew-symmetric
     * @dataProvider dataProviderForSkewSymmetricMatrix
     * @param        array $A
     * @throws       \Exception
     */
    public function testSumOfTwoSkewSymmetricMatricesIsSkewSymmetric(array $A)
    {
        // Given
        $A = MatrixFactory::create($A);

        // When
        $B = $A->add($A);

        // Then
        $this->assertTrue($B->isSkewSymmetric());
    }

    /**
     * @test Axiom: Scalar multiple of a skew-symmetric matrix is skew-symmetric
     * @dataProvider dataProviderForSkewSymmetricMatrix
     * @param        array $A
     * @throws       \Exception
     */
    public function testScalarMultipleOfSkewSymmetricMatrixIsSkewSymmetric(array $A)
    {
        $A = MatrixFactory::create($A);

        $B = $A->scalarMultiply(6);
        $this->assertTrue($B->isSkewSymmetric());
    }

    /**
     * @test         Axiom: The elements on the diagonal of a skew-symmetric matrix are zero, and therefore also its trace
     * @dataProvider dataProviderForSkewSymmetricMatrix
     * @param        array $A
     * @throws       \Exception
     */
    public function testSkewSymmetricMatrixDiagonalElementsAreZeroAndThereforeTraceIsZero(array $A)
    {
        // Given
        $A = MatrixFactory::create($A);

        // When
        foreach ($A->getDiagonalElements() as $diagonal_element) {
            // Then
            $this->assertEquals(0, $diagonal_element);
        }

        // And When
        $trace = $A->trace();

        // Then
        $this->assertEquals(0, $trace);
    }

    /**
     * @test         Axiom: A is a real skew-symmetric matrix, then I+A is invertible, where I is the identity matrix
     * @dataProvider dataProviderForSkewSymmetricMatrix
     * @param        array $A
     * @throws       \Exception
     */
    public function testSkewSymmetricMatrixAddedToIdentityIsInvertible(array $A)
    {
        // Given
        $A = MatrixFactory::create($A);
        $I = MatrixFactory::identity($A->getN());

        // When
        $I＋A = $I->add($A);

        // Then
        $this->assertTrue($I＋A->isInvertible());
    }

    /**
     * @test Axiom: A ⊗ (B + C) = A ⊗ B + A ⊗ C
     * Kronecker product bilinearity
     * @dataProvider dataProviderForThreeMatrices
     * @param        array $A
     * @param        array $B
     * @param        array $C
     * @throws       \Exception
     */
    public function testKroneckerProductBilinearity1(array $A, array $B, array $C)
    {
        // Given
        $A = MatrixFactory::create($A);
        $B = MatrixFactory::create($B);
        $C = MatrixFactory::create($C);

        // When
        $A⊗⟮B＋C⟯  = $A->kroneckerProduct($B->add($C));
        $A⊗B＋A⊗C = $A->kroneckerProduct($B)->add($A->kroneckerProduct($C));

        // Then
        $this->assertEquals($A⊗⟮B＋C⟯->getMatrix(), $A⊗B＋A⊗C->getMatrix());
    }

    /**
     * @test Axiom: (A + B) ⊗ C = A ⊗ C + B ⊗ C
     * Kronecker product bilinearity
     * @dataProvider dataProviderForThreeMatrices
     * @param        array $A
     * @param        array $B
     * @param        array $C
     * @throws       \Exception
     */
    public function testKroneckerProductBilinearity2(array $A, array $B, array $C)
    {
        // Given
        $A = MatrixFactory::create($A);
        $B = MatrixFactory::create($B);
        $C = MatrixFactory::create($C);

        // When
        $⟮A＋B⟯⊗C  = $A->add($B)->kroneckerProduct($C);
        $A⊗C＋B⊗C = $A->kroneckerProduct($C)->add($B->kroneckerProduct($C));

        // Then
        $this->assertEquals($⟮A＋B⟯⊗C->getMatrix(), $A⊗C＋B⊗C->getMatrix());
    }

    /**
     * @test Axiom: (A ⊗ B) ⊗ C = A ⊗ (B ⊗ C)
     * Kronecker product associative
     * @dataProvider dataProviderForThreeMatrices
     * @param        array $A
     * @param        array $B
     * @param        array $C
     * @throws       \Exception
     */
    public function testKroneckerProductAssociativity(array $A, array $B, array $C)
    {
        // Given
        $A = MatrixFactory::create($A);
        $B = MatrixFactory::create($B);
        $C = MatrixFactory::create($C);

        // When
        $⟮A⊗B⟯⊗C = $A->kroneckerProduct($B)->kroneckerProduct($C);
        $A⊗⟮B⊗C⟯ = $A->kroneckerProduct($B->kroneckerProduct($C));

        // Then
        $this->assertEquals($⟮A⊗B⟯⊗C->getMatrix(), $A⊗⟮B⊗C⟯->getMatrix());
    }

    /**
     * @test         Axiom: (A ⊗ B)(C ⊗ D) = AC ⊗ BD
     *               Kronecker multiplication
     * @dataProvider dataProviderForFourMatrices
     * @param        array $A
     * @param        array $B
     * @param        array $C
     * @param        array $D
     * @throws       \Exception
     */
    public function testKroneckerProductMultiplication(array $A, array $B, array $C, array $D)
    {
        // Given
        $A = MatrixFactory::create($A);
        $B = MatrixFactory::create($B);
        $C = MatrixFactory::create($C);
        $D = MatrixFactory::create($D);

        // When
        $⟮A⊗B⟯ = $A->kroneckerProduct($B);
        $⟮C⊗D⟯ = $C->kroneckerProduct($D);
        $⟮A⊗B⟯⟮C⊗D⟯ = $⟮A⊗B⟯->multiply($⟮C⊗D⟯);

        // And
        $AC = $A->multiply($C);
        $BD = $B->multiply($D);
        $AC⊗BD = $AC->kroneckerProduct($BD);

        // Theb
        $this->assertEquals($⟮A⊗B⟯⟮C⊗D⟯->getMatrix(), $AC⊗BD->getMatrix());
    }

    /**
     * @test Axiom: (kA) ⊗ B = A ⊗ (kB) = k(A ⊗ B)
     * Kronecker product scalar multiplication
     * @dataProvider dataProviderForTwoSquareMatrices
     * @param        array $A
     * @param        array $B
     * @throws       \Exception
     */
    public function testKroneckerProductScalarMultiplication(array $A, array $B)
    {
        // Given
        $A = MatrixFactory::create($A);
        $B = MatrixFactory::create($B);
        $k = 5;

        // When
        $⟮kA⟯⊗B = $A->scalarMultiply($k)->kroneckerProduct($B);
        $A⊗⟮kB⟯ = $A->kroneckerProduct($B->scalarMultiply($k));
        $k⟮A⊗B⟯ = $A->kroneckerProduct($B)->scalarMultiply($k);

        // Then
        $this->assertEquals($⟮kA⟯⊗B->getMatrix(), $A⊗⟮kB⟯->getMatrix());
        $this->assertEquals($⟮kA⟯⊗B->getMatrix(), $k⟮A⊗B⟯->getMatrix());
        $this->assertEquals($k⟮A⊗B⟯->getMatrix(), $A⊗⟮kB⟯->getMatrix());
    }

    /**
     * @test Axiom: (A ⊗ B)⁻¹ = A⁻¹ ⊗ B⁻¹
     * Inverse of Kronecker product
     * @dataProvider dataProviderForTwoSquareMatrices
     * @param        array $A
     * @param        array $B
     * @throws       \Exception
     */
    public function testKroneckerProductInverse(array $A, array $B)
    {
        // Given
        $A = MatrixFactory::create($A);
        $B = MatrixFactory::create($B);

        // When
        $A⁻¹     = $A->inverse();
        $B⁻¹     = $B->inverse();
        $A⁻¹⊗B⁻¹ = $A⁻¹->kroneckerProduct($B⁻¹);
        $⟮A⊗B⟯⁻¹  = $A->kroneckerProduct($B)->inverse();

        // Then
        $this->assertEqualsWithDelta($A⁻¹⊗B⁻¹->getMatrix(), $⟮A⊗B⟯⁻¹->getMatrix(), 0.00001);
    }

    /**
     * @test Axiom: (A ⊗ B)ᵀ = Aᵀ ⊗ Bᵀ
     * Transpose of Kronecker product
     * @dataProvider dataProviderForTwoSquareMatrices
     * @param        array $A
     * @param        array $B
     * @throws       \Exception
     */
    public function testKroneckerProductTranspose(array $A, array $B)
    {
        // Given
        $A = MatrixFactory::create($A);
        $B = MatrixFactory::create($B);

        // When
        $Aᵀ    = $A->transpose();
        $Bᵀ    = $B->transpose();
        $Aᵀ⊗Bᵀ = $Aᵀ->kroneckerProduct($Bᵀ);
        $⟮A⊗B⟯ᵀ = $A->kroneckerProduct($B)->transpose();

        // Then
        $this->assertEquals($Aᵀ⊗Bᵀ->getMatrix(), $⟮A⊗B⟯ᵀ->getMatrix());
    }

    /**
     * @test Axiom: det(A ⊗ B) = det(A)ᵐ det(B)ⁿ
     * Determinant of Kronecker product - where A is nxn matrix, and b is nxn matrix
     * @dataProvider dataProviderForKroneckerProductDeterminant
     * @param        array $A
     * @param        array $B
     * @throws       \Exception
     */
    public function testKroneckerProductDeterminant(array $A, array $B)
    {
        // Given
        $A = MatrixFactory::create($A);
        $B = MatrixFactory::create($B);

        // When
        $det⟮A⟯ᵐ  = ($A->det()) ** $B->getM();
        $det⟮B⟯ⁿ  = ($B->det()) ** $A->getN();
        $det⟮A⊗B⟯ = $A->kroneckerProduct($B)->det();

        // Then
        $this->assertEqualsWithDelta($det⟮A⊗B⟯, $det⟮A⟯ᵐ  * $det⟮B⟯ⁿ, 0.0001);
    }

    /**
     * @return array
     */
    public function dataProviderForKroneckerProductDeterminant(): array
    {
        return [
            [
                [
                    [5],
                ],
                [
                    [4],
                ],
            ],
            [
                [
                    [5, 6],
                    [2, 4],
                ],
                [
                    [4, 9],
                    [3, 1],
                ],
            ],
            [
                [
                    [5, 6],
                    [-2, 4],
                ],
                [
                    [4, -9],
                    [3, 1],
                ],
            ],
            [
                [
                    [1, 2, 3],
                    [2, 4, 6],
                    [7, 6, 5],
                ],
                [
                    [2, 3, 4],
                    [3, 1, 6],
                    [4, 3, 3],
                ],
            ],
            [
                [
                    [1, -2, 3],
                    [2, -4, 6],
                    [7, -6, 5],
                ],
                [
                    [2, 3, 4],
                    [3, 1, 6],
                    [4, 3, 3],
                ],
            ],
            [
                [
                    [1, 2, 3, 4],
                    [2, 4, 6, 8],
                    [7, 6, 5, 4],
                    [1, 3, 5, 7],
                ],
                [
                    [2, 3, 4, 5],
                    [3, 1, 6, 3],
                    [4, 3, 3, 4],
                    [3, 3, 4, 1],
                ],
            ],
            [
                [
                    [-1, 2, 3, 4],
                    [2, 4, 6, 8],
                    [7, 6, 5, 4],
                    [1, 3, 5, 7],
                ],
                [
                    [2, 3, 4, 5],
                    [3, 1, 6, 3],
                    [4, 3, 3, -4],
                    [3, 3, 4, 1],
                ],
            ],
            [
                [
                    [1, 2, 3, 4, 5],
                    [2, 4, 6, 8, 10],
                    [7, 6, 5, 4, 5],
                    [1, 3, 5, 7, 9],
                    [1, 2, 3, 4, 5],
                ],
                [
                    [2, 3, 4, 5, 2],
                    [3, 1, 6, 3, 2],
                    [4, 3, 3, 4, 2],
                    [3, 3, 4, 1, 2],
                    [1, 1, 2, 2, 1],
                ],
            ],
        ];
    }

    /**
     * @test Axiom: Kronecker sum A⊕B = A⊗Ib + I⊗aB
     * Kronecker sum is the matrix product of the Kronecker product of each matrix with the other matrix's identiry matrix.
     * @dataProvider dataProviderForTwoSquareMatrices
     * @param        array $A
     * @param        array $B
     * @throws       \Exception
     */
    public function testKroneckerSum(array $A, array $B)
    {
        // Given
        $A   = new NumericSquareMatrix($A);
        $B   = new NumericSquareMatrix($B);
        $A⊕B = $A->kroneckerSum($B);

        // When
        $In          = MatrixFactory::identity($A->getN());
        $Im          = MatrixFactory::identity($B->getM());
        $A⊗Im        = $A->kroneckerProduct($Im);
        $In⊗B        = $In->kroneckerProduct($B);
        $A⊗Im＋In⊗B = $A⊗Im->add($In⊗B);

        // Then
        $this->assertEquals($A⊕B, $A⊗Im＋In⊗B);
    }

    /**
     * @test Axiom: Covariance matrix S = Sᵀ
     * Covariance matrix is symmetric so it is the same as its transpose
     * @dataProvider dataProviderForCovarianceSymmetric
     * @param        array $A
     * @throws       \Exception
     */
    public function testCovarianceMatrixIsSymmetric(array $A)
    {
        // Given
        $A  = MatrixFactory::create($A);

        // When
        $S  = $A->covarianceMatrix();
        $Sᵀ = $S->transpose();

        // Then
        $this->assertEquals($S->getMatrix(), $Sᵀ->getMatrix());
    }

    /**
     * @return array
     */
    public function dataProviderForCovarianceSymmetric(): array
    {
        return [
            [
                [
                    [1, 4, 7, 8],
                    [2, 2, 8, 4],
                    [1, 13, 1, 5],
                ],
            ],
            [
                [
                    [19, 22, 6, 3, 2, 20],
                    [12, 6, 9, 15, 13, 5],
                ],
            ],
            [
                [
                    [4, 4.2, 3.9, 4.3, 4.1],
                    [2, 2.1, 2, 2.1, 2.2],
                    [.6, .59, .58, .62, .63]
                ],
            ],
            [
                [
                    [2.5, 0.5, 2.2, 1.9, 3.1, 2.3, 2, 1, 1.5, 1.1],
                    [2.4, 0.7, 2.9, 2.2, 3.0, 2.7, 1.6, 1.1, 1.6, 0.9],
                ],
            ],
        ];
    }

    /**
     * @test Axiom: Positive definiteness A is PD ⇔ -A is ND
     * If A is positive definite, then -A is negative definite.
     * @dataProvider dataProviderForPositiveDefiniteMatrix
     * @param        array $A
     * @throws       \Exception
     */
    public function testPositiveDefiniteNegativeisNegativeDefinite(array $A)
    {
        // Given
        $A = MatrixFactory::create($A);

        // When
        $−A = $A->scalarMultiply(-1);

        // Then
        $this->assertTrue($A->isPositiveDefinite());
        $this->assertTrue($−A->isNegativeDefinite());
    }

    /**
     * @test Axiom: Positive semidefiniteness A is PSD ⇔ -A is NSD
     * If A is positive semidefinite, then -A is negative definite.
     * @dataProvider dataProviderForPositiveSemidefiniteMatrix
     * @param        array $A
     * @throws       \Exception
     */
    public function testPositiveSemidefiniteNegativeisNegativeSemidefinite(array $A)
    {
        // Given
        $A = MatrixFactory::create($A);

        // When
        $−A = $A->scalarMultiply(-1);

        // Then
        $this->assertTrue($A->isPositiveSemidefinite());
        $this->assertTrue($−A->isNegativeSemidefinite());
    }

    /**
     * @test Axiom: Positive definiteness A is PD ⇒ A is PSD
     * If A is positive definite, then A is also positive semidefinite.
     * @dataProvider dataProviderForPositiveDefiniteMatrix
     * @param        array $A
     * @throws       \Exception
     */
    public function testPositiveDefiniteIsAlsoPositiveSemidefinite(array $A)
    {
        // Given
        $A  = MatrixFactory::create($A);

        // Then
        $this->assertTrue($A->isPositiveDefinite());
        $this->assertTrue($A->isPositiveSemidefinite());
    }

    /**
     * @test Axiom: Negative definiteness A is ND ⇒ A is NSD
     * If A is negative definite, then A is also negative semidefinite.
     * @dataProvider dataProviderForNegativeDefiniteMatrix
     * @param        array $A
     * @throws       \Exception
     */
    public function testNegativeDefiniteIsAlsoNegativeSemidefinite(array $A)
    {
        // Given
        $A  = MatrixFactory::create($A);

        // Then
        $this->assertTrue($A->isNegativeDefinite());
        $this->assertTrue($A->isNegativeSemidefinite());
    }

    /**
     * @test Axiom: Positive definiteness A is PD ⇔ A⁻¹ is PD
     * If A is positive definite, then A⁻¹ is positive definite.
     * @dataProvider dataProviderForPositiveDefiniteMatrix
     * @param        array $A
     * @throws       \Exception
     */
    public function testPositiveDefiniteInverseIsPositiveDefinite(array $A)
    {
        // Given
        $A = MatrixFactory::create($A);

        // When
        $A⁻¹ = $A->inverse();

        // Floating point adjustment
        $A⁻¹ = $A⁻¹->map(function ($x) {
            return \round($x, 7);
        });

        // Then
        $this->assertTrue($A->isPositiveDefinite());
        $this->assertTrue($A⁻¹->isPositiveDefinite());
    }

    /**
     * @test Axiom: Negative definiteness A is ND ⇔ A⁻¹ is ND
     * If A is negative definite, then A⁻¹ is negative definite.
     * @dataProvider dataProviderForNegativeDefiniteMatrix
     * @param        array $A
     * @throws       \Exception
     */
    public function testNegativeDefiniteInverseIsNegativeDefinite(array $A)
    {
        // Given
        $A = MatrixFactory::create($A);

        // When
        $A⁻¹ = $A->inverse();

        // Floating point adjustment
        $A⁻¹ = $A⁻¹->map(function ($x) {
            return \round($x, 7);
        });

        // Then
        $this->assertTrue($A->isNegativeDefinite());
        $this->assertTrue($A⁻¹->isNegativeDefinite());
    }

    /**
     * @test Axiom: Positive definiteness A is PD and r > 0 ⇒ rA is PD
     * If A is positive definite and r > 0, then rA is positive definite.
     * @dataProvider dataProviderForPositiveDefiniteMatrix
     * @param        array $A
     * @throws       \Exception
     */
    public function testPositiveDefiniteThenScalarMultiplyWithPositiveNumberIsPositiveDefinite(array $A)
    {
        // Given
        $A = MatrixFactory::create($A);
        $this->assertTrue($A->isPositiveDefinite());

        foreach (\range(1, 10) as $r) {
            // When
            $rA = $A->scalarMultiply($r);

            // Then
            $this->assertTrue($rA->isPositiveDefinite());
        }
    }

    /**
     * @test Axiom: Positive definiteness A and B are PD ⇒ A + B is PD
     * If A and B are positive definite then A + B is positive definite.
     * @dataProvider dataProviderForPositiveDefiniteMatrix
     * @param        array $M
     * @throws       \Exception
     */
    public function testPositiveDefiniteAPlusAIsPositiveDefinite(array $M)
    {
        // Given
        $A = MatrixFactory::create($M);
        $B = MatrixFactory::create($M);

        // When
        $A＋B = $A->add($B);

        // Then
        $this->assertTrue($A->isPositiveDefinite());
        $this->assertTrue($B->isPositiveDefinite());
        $this->assertTrue($A＋B->isPositiveDefinite());
    }

    /**
     * @test Axiom: Positive definiteness A and B are PD ⇒ A + B is PD
     * If A and B are positive definite then A + B is positive definite.
     * @dataProvider dataProviderForTwoPositiveDefiniteMatrices
     * @param        array $A
     * @param        array $B
     * @throws       \Exception
     */
    public function testPositiveDefiniteAPlusBIsPositiveDefinite(array $A, array $B)
    {
        // Given
        $A = MatrixFactory::create($A);
        $B = MatrixFactory::create($B);

        // When
        $A＋B = $A->add($B);

        // Then
        $this->assertTrue($A->isPositiveDefinite());
        $this->assertTrue($B->isPositiveDefinite());
        $this->assertTrue($A＋B->isPositiveDefinite());
    }

    /**
     * @test Axiom: Positive definiteness A and B are PD ⇒ ABA is PD
     * If A and B are positive definite then ABA is positive definite.
     * @dataProvider dataProviderForPositiveDefiniteMatrix
     * @param        array $M
     * @throws       \Exception
     */
    public function testPositiveDefiniteAAAIsPositiveDefinite(array $M)
    {
        // Given
        $A = MatrixFactory::create($M);
        $B = MatrixFactory::create($M);

        // When
        $ABA = $A->multiply($B)->multiply($A);

        // Then
        $this->assertTrue($A->isPositiveDefinite());
        $this->assertTrue($B->isPositiveDefinite());
        $this->assertTrue($ABA->isPositiveDefinite());
    }

    /**
     * @test Axiom: Positive definiteness A and B are PD ⇒ ABA is PD
     * If A and B are positive definite then ABA is positive definite.
     * @dataProvider dataProviderForTwoPositiveDefiniteMatrices
     * @param        array $A
     * @param        array $B
     * @throws       \Exception
     */
    public function testPositiveDefiniteABAIsPositiveDefinite(array $A, array $B)
    {
        // Given
        $A = MatrixFactory::create($A);
        $B = MatrixFactory::create($B);

        // When
        $ABA = $A->multiply($B)->multiply($A);

        // Then
        $this->assertTrue($A->isPositiveDefinite());
        $this->assertTrue($B->isPositiveDefinite());
        $this->assertTrue($ABA->isPositiveDefinite());
    }

    /**
     * @test Axiom: Positive definiteness A and B are PD ⇒ BAB is PD
     * If A and B are positive definite then BAB is positive definite.
     * @dataProvider dataProviderForTwoPositiveDefiniteMatrices
     * @param        array $A
     * @param        array $B
     * @throws       \Exception
     */
    public function testPositiveDefiniteBABIsPositiveDefinite(array $A, array $B)
    {
        // Given
        $A = MatrixFactory::create($A);
        $B = MatrixFactory::create($B);

        // When
        $BAB = $B->multiply($A)->multiply($B);

        // Then
        $this->assertTrue($A->isPositiveDefinite());
        $this->assertTrue($B->isPositiveDefinite());
        $this->assertTrue($BAB->isPositiveDefinite());
    }

    /**
     * @test Axiom: Zero matrix is lower triangular
     * @throws   \Exception
     */
    public function testZeroMatrixIsLowerTriangular()
    {
        foreach (\range(1, 20) as $m) {
            // Given
            $L = MatrixFactory::zero($m, $m);

            // Then
            $this->assertTrue($L->isLowerTriangular());
        }
    }

    /**
     * @test Axiom: Zero matrix is upper triangular
     * @throws   \Exception
     */
    public function testZeroMatrixIsUpperTriangular()
    {
        foreach (\range(1, 20) as $m) {
            // Given
            $L = MatrixFactory::zero($m, $m);

            // Then
            $this->assertTrue($L->isUpperTriangular());
        }
    }

    /**
     * @test Axiom: Zero matrix is diagonal
     * @throws   \Exception
     */
    public function testZeroMatrixIsDiagonal()
    {
        foreach (\range(1, 20) as $m) {
            // Given
            $L = MatrixFactory::zero($m, $m);

            // Then
            $this->assertTrue($L->isDiagonal());
        }
    }

    /**
     * @test Axiom: Lᵀ is upper triangular
     * Transpose of a lower triangular matrix is upper triagular
     * @dataProvider dataProviderForLowerTriangularMatrix
     * @param        array $L
     * @throws       \Exception
     */
    public function testTransposeOfLowerTriangularMatrixIsUpperTriangular(array $L)
    {
        // Given
        $L = MatrixFactory::create($L);

        // When
        $Lᵀ = $L->Transpose();

        // Then
        $this->assertTrue($L->isLowerTriangular());
        $this->assertTrue($Lᵀ->isUpperTriangular());
    }

    /**
     * @test Axiom: Uᵀ is lower triangular
     * Transpose of an upper triangular matrix is lower triagular
     * @dataProvider dataProviderForUpperTriangularMatrix
     * @param        array $U
     * @throws       \Exception
     */
    public function testTransposeOfUpperTriangularMatrixIsLowerTriangular(array $U)
    {
        // Given
        $U = MatrixFactory::create($U);

        // When
        $Uᵀ = $U->Transpose();

        // Then
        $this->assertTrue($U->isUpperTriangular());
        $this->assertTrue($Uᵀ->isLowerTriangular());
    }

    /**
     * @test Axiom: LL is lower triangular
     * Product of two lower triangular matrices is lower triangular
     * @dataProvider dataProviderForLowerTriangularMatrix
     * @param        array $L
     * @throws       \Exception
     */
    public function testProductOfTwoLowerTriangularMatricesIsLowerTriangular(array $L)
    {
        // Given
        $L = MatrixFactory::create($L);

        // When
        $LL = $L->multiply($L);

        // Then
        $this->assertTrue($L->isLowerTriangular());
        $this->assertTrue($LL->isLowerTriangular());
    }

    /**
     * @test Axiom: UU is upper triangular
     * Product of two upper triangular matrices is upper triangular
     * @dataProvider dataProviderForUpperTriangularMatrix
     * @param        array $U
     * @throws       \Exception
     */
    public function testProductOfTwoUpperTriangularMatricesIsUpperTriangular(array $U)
    {
        // Given
        $U = MatrixFactory::create($U);

        // When
        $UU = $U->multiply($U);

        // Then
        $this->assertTrue($U->isUpperTriangular());
        $this->assertTrue($UU->isUpperTriangular());
    }

    /**
     * @test Axiom: L + L is lower triangular
     * Sum of two lower triangular matrices is lower triangular
     * @dataProvider dataProviderForLowerTriangularMatrix
     * @param        array $L
     * @throws       \Exception
     */
    public function testSumOfTwoLowerTriangularMatricesIsLowerTriangular(array $L)
    {
        // Given
        $L = MatrixFactory::create($L);

        // When
        $L＋L = $L->add($L);

        // Then
        $this->assertTrue($L->isLowerTriangular());
        $this->assertTrue($L＋L->isLowerTriangular());
    }

    /**
     * @test Axiom: U + U is upper triangular
     * Sum of two upper triangular matrices is upper triangular
     * @dataProvider dataProviderForUpperTriangularMatrix
     * @param        array $U
     * @throws       \Exception
     */
    public function testSumOfTwoUpperTriangularMatricesIsUpperTriangular(array $U)
    {
        // Given
        $U = MatrixFactory::create($U);

        // When
        $U＋U = $U->add($U);

        // Then
        $this->assertTrue($U->isUpperTriangular());
        $this->assertTrue($U＋U->isUpperTriangular());
    }

    /**
     * @test Axiom: L⁻¹ is lower triangular (If L is invertible)
     * The inverse of an invertible lower triangular matrix is lower triangular
     * @dataProvider dataProviderForLowerTriangularMatrix
     * @param        array $L
     * @throws       \Exception
     */
    public function testInverseOfInvertibleLowerTriangularMatrixIsLowerTriangular(array $L)
    {
        // Given
        $L = MatrixFactory::create($L);
        $this->assertTrue($L->isLowerTriangular());

        if ($L->isInvertible()) {
            // When
            $L⁻¹ = $L->inverse();

            // Then
            $this->assertTrue($L⁻¹->isLowerTriangular());
        }
    }

    /**
     * @test Axiom: U⁻¹ is upper triangular (If U is invertible)
     * The inverse of an invertible upper triangular matrix is upper triangular
     * @dataProvider dataProviderForUpperTriangularMatrix
     * @param        array $U
     * @throws       \Exception
     */
    public function testInverseOfInvertibleUpperTriangularMatrixIsUpperTriangular(array $U)
    {
        // Given
        $U = MatrixFactory::create($U);
        $this->assertTrue($U->isUpperTriangular());

        if ($U->isInvertible()) {
            // When
            $U⁻¹ = $U->inverse();

            // Then
            $this->assertTrue($U⁻¹->isUpperTriangular());
        }
    }

    /**
     * @test Axiom: kL is lower triangular
     * Product of a lower triangular matrix by a constant is lower triangular
     * @dataProvider dataProviderForLowerTriangularMatrix
     * @param        array $L
     * @throws       \Exception
     */
    public function testProductOfLowerTriangularMatrixByConstantIsLowerTriangular(array $L)
    {
        // Given
        $L = MatrixFactory::create($L);
        $this->assertTrue($L->isLowerTriangular());

        foreach (\range(1, 10) as $k) {
            // When
            $kL = $L->scalarMultiply($k);

            // Then
            $this->assertTrue($kL->isLowerTriangular());
        }
    }

    /**
     * @test Axiom: kU is upper triangular
     * Product of a upper triangular matrix by a constant is upper triangular
     * @dataProvider dataProviderForUpperTriangularMatrix
     * @param        array $U
     * @throws       \Exception
     */
    public function testProductOfUpperTriangularMatrixByConstantIsUpperTriangular(array $U)
    {
        // Given
        $U = MatrixFactory::create($U);
        $this->assertTrue($U->isUpperTriangular());

        foreach (\range(1, 10) as $k) {
            // When
            $kU = $U->scalarMultiply($k);

            // Then
            $this->assertTrue($kU->isUpperTriangular());
        }
    }

    /**
     * @test Axiom: L is invertible iff diagonal is all non zero
     * Lower triangular matrix is invertible if and only if its diagonal entries are all non zero
     * @dataProvider dataProviderForLowerTriangularMatrix
     * @param        array $L
     * @throws       \Exception
     */
    public function testLowerTriangularMatrixIsInvertibleIfAndOnlyIfDigaonalEntriesAreAllNonZero(array $L)
    {
        // Given
        $L = MatrixFactory::create($L);
        $this->assertTrue($L->isLowerTriangular());

        $zeros = \array_filter(
            $L->getDiagonalElements(),
            function ($x) {
                return $x == 0;
            }
        );

        // Then
        if (count($zeros) == 0) {
            $this->assertTrue($L->isInvertible());
        } else {
            $this->assertFalse($L->isInvertible());
        }
    }

    /**
     * @test Axiom: U is invertible iff diagonal is all non zero
     * Upper triangular matrix is invertible if and only if its diagonal entries are all non zero
     * @dataProvider dataProviderForUpperTriangularMatrix
     * @param        array $U
     * @throws       \Exception
     */
    public function testUpperTriangularMatrixIsInvertibleIfAndOnlyIfDigaonalEntriesAreAllNonZero(array $U)
    {
        // Given
        $U = MatrixFactory::create($U);
        $this->assertTrue($U->isUpperTriangular());

        $zeros = \array_filter(
            $U->getDiagonalElements(),
            function ($x) {
                return $x == 0;
            }
        );

        // Then
        if (count($zeros) == 0) {
            $this->assertTrue($U->isInvertible());
        } else {
            $this->assertFalse($U->isInvertible());
        }
    }

    /**
     * @test Axiom: Dᵀ is diagonal
     * Transpose of a diagonal matrix is diagonal
     * @dataProvider dataProviderForDiagonalMatrix
     * @param        array $D
     * @throws       \Exception
     */
    public function testTransposeOfDiagonalMatrixIsDiagonal(array $D)
    {
        // Given
        $D = MatrixFactory::create($D);

        // When
        $Dᵀ = $D->Transpose();

        // Then
        $this->assertTrue($D->isDiagonal());
        $this->assertTrue($Dᵀ->isDiagonal());
    }

    /**
     * @test Axiom: DD is diagonal
     * Product of two diagonal matrices is diagonal
     * @dataProvider dataProviderForDiagonalMatrix
     * @param        array $D
     * @throws       \Exception
     */
    public function testProductOfTwoDiagonalMatricesIsDiagonal(array $D)
    {
        // Given
        $D = MatrixFactory::create($D);

        // When
        $DD = $D->multiply($D);

        // Then
        $this->assertTrue($D->isDiagonal());
        $this->assertTrue($DD->isDiagonal());
    }

    /**
     * @test Axiom: D + D is diagonal
     * Sum of two diagonal matrices is diagonal
     * @dataProvider dataProviderForDiagonalMatrix
     * @param        array $D
     * @throws       \Exception
     */
    public function testSumOfTwoDiagonalMatricesIsDiagonal(array $D)
    {
        // Given
        $D = MatrixFactory::create($D);

        // When
        $D＋D = $D->add($D);

        // Then
        $this->assertTrue($D->isDiagonal());
        $this->assertTrue($D＋D->isDiagonal());
    }

    /**
     * @test Axiom: D⁻¹ is diagonal (If D is invertible)
     * The inverse of an invertible diagonal matrix is diagonal
     * @dataProvider dataProviderForDiagonalMatrix
     * @param        array $D
     * @throws       \Exception
     */
    public function testInverseOfInvertibleDiagonalMatrixIsDiagonal(array $D)
    {
        // Given
        $D = MatrixFactory::create($D);
        $this->assertTrue($D->isDiagonal());

        if ($D->isInvertible()) {
            // When
            $D⁻¹ = $D->inverse();
            // Then
            $this->assertTrue($D⁻¹->isDiagonal());
        }
    }

    /**
     * @test Axiom: kD is Diagonal
     * Product of a diagonal matrix by a constant is diagonal
     * @dataProvider dataProviderForDiagonalMatrix
     * @param        array $D
     * @throws       \Exception
     */
    public function testProductOfDiagonalMatrixByConstantIsDiagonal(array $D)
    {
        // Given
        $D = MatrixFactory::create($D);
        $this->assertTrue($D->isDiagonal());

        foreach (\range(1, 10) as $k) {
            // When
            $kD = $D->scalarMultiply($k);
            // Then
            $this->assertTrue($kD->isDiagonal());
        }
    }

    /**
     * @test Axiom: D is invertible iff diagonal is all non zero
     * Diagonal matrix is invertible if and only if its diagonal entries are all non zero
     * @dataProvider dataProviderForDiagonalMatrix
     * @param        array $D
     * @throws       \Exception
     */
    public function testDiagonalMatrixIsInvertibleIfAndOnlyIfDigaonalEntriesAreAllNonZero(array $D)
    {
        // Given
        $D = MatrixFactory::create($D);
        $this->assertTrue($D->isDiagonal());

        $zeros = \array_filter(
            $D->getDiagonalElements(),
            function ($x) {
                return $x == 0;
            }
        );

        // Then
        if (count($zeros) == 0) {
            $this->assertTrue($D->isInvertible());
        } else {
            $this->assertFalse($D->isInvertible());
        }
    }

    /**
     * @test Axiom: Reduced row echelon form is upper triangular
     * @dataProvider dataProviderForSquareMatrix
     * @param        array $A
     * @throws       \Exception
     */
    public function testReducedRowEchelonFormIsUpperTriangular(array $A)
    {
        // Given
        $A = MatrixFactory::create($A);

        // When
        $rref = $A->rref();

        // Then
        $this->assertTrue($rref->isUpperTriangular());
    }

    /**
     * @test Axiom: Jᵀ = J
     * Transpose of an exchange matrix is itself
     * @throws \Exception
     */
    public function testTransposeOfExchangeMatrix()
    {
        foreach (\range(1, 20) as $n) {
            // Given
            $J  = MatrixFactory::exchange($n);
            // When
            $Jᵀ = $J->transpose();
            // Then
            $this->assertEquals($J->getMatrix(), $Jᵀ->getMatrix());
        }
    }

    /**
     * @test Axiom: J⁻¹ = J
     * Inverse of an exchange matrix is itself
     * @throws \Exception
     */
    public function testInverseOfExchangeMatrix()
    {
        foreach (\range(1, 20) as $n) {
            // Given
            $J  = MatrixFactory::exchange($n);
            // When
            $J⁻¹ = $J->inverse();
            // Then
            $this->assertEquals($J->getMatrix(), $J⁻¹->getMatrix());
        }
    }

    /**
     * @test Axiom: tr(J) is 1 if n is odd, and 0 if n is even
     * Trace of J is 1 if n is odd, and 0 is n is even.
     * @throws \Exception
     */
    public function testTraceOfExchangeMatrix()
    {
        foreach (\range(1, 20) as $n) {
            // Given
            $J    = MatrixFactory::exchange($n);
            // When
            $tr⟮J⟯ = $J->trace();

            // Then
            if (Integer::isOdd($n)) {
                $this->assertEquals(1, $tr⟮J⟯);
            } else {
                $this->assertEquals(0, $tr⟮J⟯);
            }
        }
    }

    /**
     * @test Axiom: Signature matrix is involutory
     * @dataProvider dataProviderForSignatureMatrix
     * @param        array $A
     * @throws       \Exception
     */
    public function testSignatureMatrixIsInvolutory(array $A)
    {
        // Given
        $A = MatrixFactory::create($A);

        // Then
        $this->assertTrue($A->isSignature());
        $this->assertTrue($A->isInvolutory());
    }

    /**
     * @test Axiom: Hilbert matrix is symmetric
     * @throws   \Exception
     */
    public function testHilbertMatrixIsSymmetric()
    {
        foreach (\range(1, 10) as $n) {
            // Given
            $H = MatrixFactory::hilbert($n);
            // Then
            $this->assertTrue($H->isSymmetric());
        }
    }

    /**
     * @test Axiom: Hilbert matrix is positive definite
     * @throws   \Exception
     */
    public function testHilbertMatrixIsPositiveDefinite()
    {
        foreach (\range(1, 10) as $n) {
            // Given
            $H = MatrixFactory::hilbert($n);
            // Then
            $this->assertTrue($H->isPositiveDefinite());
        }
    }

    /**
     * @test         Axiom: A = LLᵀ (Cholesky decomposition)
     * @dataProvider dataProviderForPositiveDefiniteMatrix
     * @param        array $A
     * @throws       \Exception
     */
    public function testCholeskyDecompositionLTimesLTransposeIsA(array $A)
    {
        // Given
        $A        = MatrixFactory::create($A);
        $cholesky = $A->choleskyDecomposition();
        $L        = $cholesky->L;
        $Lᵀ       = $cholesky->LT;

        // When
        $LLᵀ = $L->multiply($Lᵀ);

        // Then
        $this->assertEqualsWithDelta($A->getMatrix(), $LLᵀ->getMatrix(), 0.00001);
    }

    /**
     * @test         Axiom: L is lower triangular (Cholesky decomposition)
     * @dataProvider dataProviderForPositiveDefiniteMatrix
     * @param        array $A
     * @throws       \Exception
     */
    public function testCholeskyDecompositionLIsLowerTriangular(array $A)
    {
        // Given
        $A = MatrixFactory::create($A);

        // When
        $cholesky = $A->choleskyDecomposition();

        // Then
        $this->assertTrue($cholesky->L->isLowerTriangular());
    }

    /**
     * @test         Axiom: Lᵀ is upper triangular (Cholesky decomposition)
     * @dataProvider dataProviderForPositiveDefiniteMatrix
     * @param        array $A
     * @throws       \Exception
     */
    public function testCholeskyDecompositionLTransposeIsUpperTriangular(array $A)
    {
        // Given
        $A = MatrixFactory::create($A);

        // When
        $cholesky = $A->choleskyDecomposition();

        // Then
        $this->assertTrue($cholesky->LT->isUpperTriangular());
    }

    /**
     * @test         Axiom: adj⟮A⟯ = Cᵀ
     *               Adjugate matrix equals the transpose of the cofactor matrix
     * @dataProvider dataProviderForSquareMatrixGreaterThanOne
     * @param        array $A
     * @throws       \Exception
     */
    public function testAdjugateIsTransoseOfCofactorMatrix(array $A)
    {
        // Given
        $A = MatrixFactory::create($A);

        // When
        $adj⟮A⟯ = $A->adjugate();
        $Cᵀ    = $A->cofactorMatrix()->transpose();

        // Then
        $this->assertEqualsWithDelta($adj⟮A⟯, $Cᵀ, 0.00001);
    }

    /**
     * @test         Axiom: A adj⟮A⟯ = det⟮A⟯ I
     *               The product of A with its adjugate yields a diagonal matrix whose diagonal entries are det(A)
     * @dataProvider dataProviderForSquareMatrixGreaterThanOne
     * @param        array $A
     * @throws       \Exception
     */
    public function testAdjugateTimesAIsIdentityMatrixTimesDeterminantOfA(array $A)
    {
        // Given
        $A = MatrixFactory::create($A);

        // When
        $adj⟮A⟯ = $A->adjugate();
        $Aadj⟮A⟯ = $A->multiply($adj⟮A⟯);

        // And
        $I     = MatrixFactory::identity($A->getN());
        $det⟮A⟯ = $A->det();
        $det⟮A⟯I = $I->scalarMultiply($det⟮A⟯);

        // Then
        $this->assertEqualsWithDelta($Aadj⟮A⟯, $det⟮A⟯I, 0.00001);
    }

    /**
     * @test         Axiom: adj⟮A⟯ = det⟮A⟯A⁻¹
     *               The product of A with its adjugate yields a diagonal matrix whose diagonal entries are det(A)
     * @dataProvider dataProviderForNonsingularMatrix
     * @param        array $A
     * @throws       \Exception
     */
    public function testAdjugateEqualsInverseOfATimesDeterminant(array $A)
    {
        // Given
        $A = MatrixFactory::create($A);

        // When
        $A⁻¹   = $A->inverse();
        $adj⟮A⟯ = $A->adjugate();
        $det⟮A⟯ = $A->det();
        $det⟮A⟯A⁻¹ = $A⁻¹->scalarMultiply($det⟮A⟯);

        // Then
        $this->assertEqualsWithDelta($adj⟮A⟯, $det⟮A⟯A⁻¹, 0.00001);
    }

    /**
     * @test         Axiom: A⁻¹ = (1/det⟮A⟯) adj⟮A⟯
     *               The inverse of a matrix is equals to one over the determinant multiplied by the adjugate
     * @dataProvider dataProviderForNonsingularMatrix
     * @param        array $A
     * @throws       \Exception
     */
    public function testInverseEqualsOneOverDetTimesAdjugate(array $A)
    {
        // Given
        $A = MatrixFactory::create($A);

        // When
        $A⁻¹           = $A->inverse();
        $adj⟮A⟯         = $A->adjugate();
        $det⟮A⟯         = $A->det();
        $⟮1／det⟮A⟯⟯adj⟮A⟯ = $adj⟮A⟯->scalarMultiply(1 / $det⟮A⟯);

        // Then
        $this->assertEqualsWithDelta($A⁻¹, $⟮1／det⟮A⟯⟯adj⟮A⟯, 0.00001);
    }

    /**
     * @test         Axiom: adj⟮I⟯ = I
     *               The adjugate of identity matrix is identity matrix
     * @dataProvider dataProviderForIdentityMatrix
     * @param        array $I
     * @throws       \Exception
     */
    public function testAdjugateOfIdenetityMatrixIsIdentity(array $I)
    {
        // Given
        $I = MatrixFactory::create($I);

        // When
        $adj⟮I⟯ = $I->adjugate();

        // Then
        $this->assertEqualsWithDelta($adj⟮I⟯, $I, 0.00001);
    }

    /**
     * @test         Axiom: adj⟮AB⟯ = adj⟮B⟯adj⟮A⟯
     *               The adjugate of AB equals the adjugate of B times the adjugate of A
     * @dataProvider dataProviderForTwoNonsingularMatrices
     * @param        array $A
     * @param        array $B
     * @throws       \Exception
     */
    public function testAdjugateABEqualsAdjugateBTimesAdjugateA(array $A, array $B)
    {
        // Given
        $A = MatrixFactory::create($A);
        $B = MatrixFactory::create($B);

        // When
        $AB     = $A->multiply($B);
        $adj⟮A⟯  = $A->adjugate();
        $adj⟮B⟯  = $B->adjugate();
        $adj⟮AB⟯ = $AB->adjugate();
        $adj⟮B⟯adj⟮A⟯ = $adj⟮B⟯->multiply($adj⟮A⟯);

        // Then
        $this->assertEqualsWithDelta($adj⟮AB⟯, $adj⟮B⟯adj⟮A⟯, 0.00001);
    }

    /**
     * @test         Axiom: adj⟮cA⟯ = cⁿ⁻¹ adj⟮A⟯
     *               The adjugate of a matrix times a scalar equals the adjugate of the matrix then times a scalar raised to n - 1
     * @dataProvider dataProviderForNonsingularMatrix
     * @param        array $A
     * @throws       \Exception
     */
    public function testAdjugateAtimesCEqualsAdjugateATimesCRaisedToNMinusOne(array $A)
    {
        // Given
        $c = 4;
        $A = MatrixFactory::create($A);

        // When
        $cA        = $A->scalarMultiply($c);
        $adj⟮A⟯     = $A->adjugate();
        $adj⟮cA⟯    = $cA->adjugate();
        $cⁿ⁻¹      = \pow($c, $A->getN() - 1);
        $cⁿ⁻¹adj⟮A⟯ = $adj⟮A⟯->scalarMultiply($cⁿ⁻¹);

        // Then
        $this->assertEqualsWithDelta($adj⟮cA⟯, $cⁿ⁻¹adj⟮A⟯, 0.00001);
    }

    /**
     * @test         Axiom: adj⟮B⟯adj⟮A⟯ = det⟮B⟯B⁻¹ det⟮A⟯A⁻¹ = det⟮AB⟯⟮AB⟯⁻¹
     *               The adjugate of B times adjugate A equals the determinant of B times inverse of B times determinant of A times inverse of A
     *               which equals the determinant of AB times the inverse of AB
     * @dataProvider dataProviderForTwoNonsingularMatrices
     * @param        array $A
     * @param        array $B
     * @throws       \Exception
     */
    public function testAdjugateBTimesAdjugateAEqualsDetBTimesInverseBTimesDetATimesInverseAEqualsDetABTimesInverseAB(array $A, array $B)
    {
        // Given
        $A      = MatrixFactory::create($A);
        $B      = MatrixFactory::create($B);
        $A⁻¹    = $A->inverse();
        $B⁻¹    = $B->inverse();
        $AB     = $A->multiply($B);
        $⟮AB⟯⁻¹  = $AB->inverse();
        $adj⟮A⟯  = $A->adjugate();
        $adj⟮B⟯  = $B->adjugate();
        $det⟮A⟯  = $A->det();
        $det⟮B⟯  = $B->det();
        $det⟮AB⟯ = $AB->det();

        // When
        $det⟮A⟯A⁻¹ = $A⁻¹->scalarMultiply($det⟮A⟯);
        $det⟮B⟯B⁻¹ = $B⁻¹->scalarMultiply($det⟮B⟯);

        // And
        $adj⟮B⟯adj⟮A⟯       = $adj⟮B⟯->multiply($adj⟮A⟯);
        $det⟮B⟯B⁻¹det⟮A⟯A⁻¹ = $det⟮B⟯B⁻¹->multiply($det⟮A⟯A⁻¹);
        $det⟮AB⟯⟮AB⟯⁻¹      = $⟮AB⟯⁻¹->scalarMultiply($det⟮AB⟯);

        // Then
        $this->assertEqualsWithDelta($adj⟮B⟯adj⟮A⟯, $det⟮B⟯B⁻¹det⟮A⟯A⁻¹, 0.001);
        $this->assertEqualsWithDelta($det⟮B⟯B⁻¹det⟮A⟯A⁻¹, $det⟮AB⟯⟮AB⟯⁻¹, 0.001);
        $this->assertEqualsWithDelta($adj⟮B⟯adj⟮A⟯, $det⟮AB⟯⟮AB⟯⁻¹, 0.001);
    }

    /**
     * @test         Axiom: adj⟮Aᵀ⟯ = adj⟮A⟯ᵀ
     *               The adjugate of a matrix transpase equals the transpose of a matrix adjugate
     * @dataProvider dataProviderForNonsingularMatrix
     * @param        array $A
     * @throws       \Exception
     */
    public function testAdjugateOfTransposeEqualsTransposeOfAdjugate(array $A)
    {
        // Given
        $A = MatrixFactory::create($A);

        // When
        $Aᵀ     = $A->transpose();
        $adj⟮A⟯  = $A->adjugate();
        $adj⟮Aᵀ⟯ = $Aᵀ->adjugate();
        $adj⟮A⟯ᵀ = $adj⟮A⟯->transpose();

        // Then
        $this->assertEqualsWithDelta($adj⟮Aᵀ⟯, $adj⟮A⟯ᵀ, 0.00001);
    }

    /**
     * @test         Axiom: Aadj⟮A⟯ = adj⟮A⟯A = det⟮A⟯I
     *               A matrix times its adjugate equals the adjugate times the matrix which equals the identity matrix times the determinant
     * @dataProvider dataProviderForNonsingularMatrix
     * @param        array $A
     * @throws       \Exception
     */
    public function testMatrixTimesItsAdjugateEqualsAdjugateTimesMatrixEqualsDetTimesIdentity(array $A)
    {
        // Given
        $A = MatrixFactory::create($A);

        // When
        $adj⟮A⟯  = $A->adjugate();
        $Aadj⟮A⟯ = $A->multiply($adj⟮A⟯);
        $adj⟮A⟯A = $adj⟮A⟯->multiply($A);
        $det⟮A⟯  = $A->det();
        $I      = MatrixFactory::identity($A->getN());
        $det⟮A⟯I = $I->scalarMultiply($det⟮A⟯);

        // Then
        $this->assertEqualsWithDelta($Aadj⟮A⟯, $adj⟮A⟯A, 0.0001);
        $this->assertEqualsWithDelta($Aadj⟮A⟯, $det⟮A⟯I, 0.0001);
        $this->assertEqualsWithDelta($adj⟮A⟯A, $det⟮A⟯I, 0.0001);
    }

    /**
     * @test         Axiom: rank(A) ≤ min(m, n)
     *               The rank of a matrix is less than or equal to the minimum dimension of the matrix
     * @dataProvider dataProviderForSingleMatrix
     * @param        array $A
     * @throws       \Exception
     */
    public function testRankLessThanMinDimension(array $A)
    {
        // Given
        $A = MatrixFactory::create($A);

        // Then
        $this->assertLessThanOrEqual(\min($A->getM(), $A->getN()), $A->rank());
    }

    /**
     * @test Axiom: Zero matrix has rank of 0
     * @throws   \Exception
     */
    public function testZeroMatrixHasRankOfZero()
    {
        foreach (\range(1, 10) as $m) {
            foreach (\range(1, 10) as $n) {
                // Given
                $A = MatrixFactory::zero($m, $n);
                // Then
                $this->assertEquals(0, $A->rank());
            }
        }
    }

    /**
     * @test         Axiom: If A is square matrix, then it is invertible only if rank = n (full rank)
     * @dataProvider dataProviderForSquareMatrix
     * @param        array $A
     * @throws       \Exception
     */
    public function testSquareMatrixInvertibleIfFullRank(array $A)
    {
        // Given
        $A = MatrixFactory::create($A);

        // When
        $rank = $A->rank();

        // Then
        if ($rank === $A->getM()) {
            $this->assertTrue($A->isInvertible());
        } else {
            $this->assertFalse($A->isInvertible());
        }
    }

    /**
     * @test         Axiom: rank(AᵀA) = rank(AAᵀ) = rank(A) = rank(Aᵀ)
     * @dataProvider dataProviderForSingleMatrix
     * @param        array $A
     * @throws       \Exception
     */
    public function testRankTransposeEqualities(array $A)
    {
        // Given
        $A   = MatrixFactory::create($A);
        $Aᵀ  = $A->transpose();
        $AᵀA = $Aᵀ->multiply($A);
        $AAᵀ = $A->multiply($Aᵀ);

        // When
        $rank⟮A⟯   = $A->rank();
        $rank⟮Aᵀ⟯  = $Aᵀ->rank();
        $rank⟮AᵀA⟯ = $AᵀA->rank();
        $rank⟮AAᵀ⟯ = $AAᵀ->rank();

        // Then
        $this->assertEquals($rank⟮A⟯, $rank⟮Aᵀ⟯);
        $this->assertEquals($rank⟮A⟯, $rank⟮AᵀA⟯);
        $this->assertEquals($rank⟮A⟯, $rank⟮AAᵀ⟯);
        $this->assertEquals($rank⟮Aᵀ⟯, $rank⟮AᵀA⟯);
        $this->assertEquals($rank⟮Aᵀ⟯, $rank⟮AAᵀ⟯);
        $this->assertEquals($rank⟮AᵀA⟯, $rank⟮AAᵀ⟯);
    }

    /**
     * @test         Axiom: Lower bidiagonal matrix is upper Hessenberg
     * @dataProvider dataProviderForLowerBidiagonalMatrix
     * @param        array $A
     * @throws       \Exception
     */
    public function testLowerBidiagonalMatrixIsUpperHessenberg(array $A)
    {
        // Given
        $A = MatrixFactory::create($A);

        // Then
        $this->assertTrue($A->isLowerBidiagonal());
        $this->assertTrue($A->isUpperHessenberg());
    }

    /**
     * @test         Axiom: Upper bidiagonal matrix is lower Hessenberg
     * @dataProvider dataProviderForUpperBidiagonalMatrix
     * @param        array $A
     * @throws       \Exception
     */
    public function testUpperBidiagonalMatrixIsLowerHessenberg(array $A)
    {
        // Given
        $A = MatrixFactory::create($A);

        // Then
        $this->assertTrue($A->isUpperBidiagonal());
        $this->assertTrue($A->isLowerHessenberg());
    }

    /**
     * @test         Axiom: A matrix that is both upper Hessenberg and lower Hessenberg is a tridiagonal matrix
     * @dataProvider dataProviderForTridiagonalMatrix
     * @param        array $A
     * @throws       \Exception
     */
    public function testTridiagonalMatrixIsUpperAndLowerHessenberg(array $A)
    {
        // Given
        $A = MatrixFactory::create($A);

        // Then
        $this->assertTrue($A->isTridiagonal());
        $this->assertTrue($A->isUpperHessenberg());
        $this->assertTrue($A->isLowerHessenberg());
    }

    /**
     * @test         Axiom: AAᵀ = I for an orthogonal matrix A
     * @dataProvider dataProviderForOrthogonalMatrix
     * @param        array $A
     * @throws       \Exception
     */
    public function testOrthogonalMatrixTimesTransposeIsIdentityMatrix(array $A)
    {
        // Given
        $A  = MatrixFactory::create($A);
        $Aᵀ = $A->transpose();
        $I  = MatrixFactory::identity($A->getM());

        // When
        $AAᵀ = $A->multiply($Aᵀ);

        // Then
        $this->assertEqualsWithDelta($I->getMatrix(), $AAᵀ->getMatrix(), 0.00001);
    }

    /**
     * @test         Axiom: AᵀA = I for an orthogonal matrix A
     * @dataProvider dataProviderForOrthogonalMatrix
     * @param        array $A
     * @throws       \Exception
     */
    public function testOrthogonalTransposeOfOrthogonalMatrixTimesMatrixIsIdentityMatrix(array $A)
    {
        // Given
        $A  = MatrixFactory::create($A);
        $Aᵀ = $A->transpose();
        $I  = MatrixFactory::identity($A->getM());

        // When
        $AᵀA = $Aᵀ->multiply($A);

        // Then
        $this->assertEqualsWithDelta($I->getMatrix(), $AᵀA->getMatrix(), 0.00001);
    }

    /**
     * @test         Axiom: A⁻¹ = Aᵀ for an orthogonal matrix A
     * @dataProvider dataProviderForOrthogonalMatrix
     * @param        array $A
     * @throws       \Exception
     */
    public function testOrthogonalMatrixInverseEqualsTransposeOfOrthogonalMatrix(array $A)
    {
        // Given
        $A = MatrixFactory::create($A);

        // When
        $Aᵀ  = $A->transpose();
        $A⁻¹ = $A->inverse();

        // Then
        $this->assertEqualsWithDelta($A⁻¹->getMatrix(), $Aᵀ->getMatrix(), 0.00001);
    }

    /**
     * @test         Axiom: det(A) = ±1 for an orthogonal matrix A
     * @dataProvider dataProviderForOrthogonalMatrix
     * @param        array $A
     * @throws       \Exception
     */
    public function testOrthogonalMatrixDeterminateIsOne(array $A)
    {
        // Given
        $A = MatrixFactory::create($A);

        // When
        $det⟮A⟯ = $A->det();

        // Then
        $this->assertEqualsWithDelta(1, \abs($det⟮A⟯), 0.000001);
    }

    /**
     * @test         Axiom: Householder transformation creates a matrix that is involutory
     * @dataProvider dataProviderForSquareMatrix
     * @param        array $A
     * @throws       \Exception
     */
    public function testHouseholderTransformMatrixInvolutoryProperty(array $A)
    {
        // Given
        $A = MatrixFactory::create($A);

        // When
        $H = $A->householder();

        // Then
        $this->assertTrue($H->isInvolutory());
    }

    /**
     * @test         Axiom: Householder transformation creates a matrix with a determinant that is -1
     * @dataProvider dataProviderForSquareMatrix
     * @param        array $A
     * @throws       \Exception
     */
    public function testHouseholderTransformMatrixDeterminant(array $A)
    {
        // Given
        $A = MatrixFactory::create($A);

        // When
        $H = $A->householder();

        // Then
        $this->assertEqualsWithDelta(-1, $H->det(), 0.000001);
    }

    /**
     * @test         Axiom: Householder transformation creates a matrix that has eigenvalues 1 and -1
     * @dataProvider dataProviderForSquareMatrixGreaterThanOne
     * @param        array $A
     * @throws       \Exception
     */
    public function testHouseholderTransformMatrixEigenvalues(array $A)
    {
        // Given
        $A = MatrixFactory::create($A);

        // When
        $H = $A->householder();

        // Then
        $eigenvalues = \array_filter(
            $H->eigenvalues(),
            function ($x) {
                return !\is_nan($x);
            }
        );
        $this->assertEqualsWithDelta(1, max($eigenvalues), 0.00001);
        $this->assertEqualsWithDelta(-1, \min($eigenvalues), 0.00001);
    }

    /**
     * @test         Axiom: Nilpotent matrix - tr(Aᵏ) = 0 for all k > 0
     * @dataProvider dataProviderForNilpotentMatrix
     * @param        array $A
     * @throws       \Exception
     */
    public function testNilpotentTraceIsZero(array $A)
    {
        // Given
        $A = MatrixFactory::create($A);

        foreach (\range(1, 5) as $_) {
            // When
            $A     = $A->multiply($A);
            $trace = $A->trace();

            // Then
            $this->assertEquals(0, $trace);
        }
    }

    /**
     * @test         Axiom: Nilpotent matrix - det = 0
     * @dataProvider dataProviderForNilpotentMatrix
     * @param        array $A
     * @throws       \Exception
     */
    public function testNilpotentDetIsZero(array $A)
    {
        // Given
        $A = MatrixFactory::create($A);

        // When
        $det = $A->det();

        // Then
        $this->assertEquals(0, $det);
    }

    /**
     * @test         Axiom: Nilpotent matrix - Cannot be invertible
     * @dataProvider dataProviderForNilpotentMatrix
     * @param        array $A
     * @throws       \Exception
     */
    public function testNilpotentCannotBeInvertible(array $A)
    {
        // Given
        $A = MatrixFactory::create($A);

        // When
        $isInvertible = $A->isInvertible();

        // Then
        $this->assertFalse($isInvertible);
    }
}