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Add eigen axioms unit tests.
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‎tests/LinearAlgebra/Eigen/EigenAxiomsTest.php‎

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<?php
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namespace MathPHP\Tests\LinearAlgebra\Eigen;
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use MathPHP\LinearAlgebra\MatrixFactory;
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use MathPHP\LinearAlgebra\Vector;
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use MathPHP\Tests;
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class EigenAxiomsTest extends \PHPUnit\Framework\TestCase
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{
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use Tests\LinearAlgebra\Fixture\MatrixDataProvider;
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/**
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* Tests of Matrix eigenvector and eigenvalue axioms
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* These tests don't test specific functions,
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* but rather matrix axioms which in term make use of multiple functions.
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* If all the Matrix math is implemented properly, these tests should
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* all work out according to the axioms.
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*
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* Axioms tested:
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* - Eigenvalues and eigenvectors
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* - Av = λv (basic eigenvalue equation)
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* - det(A - λI) = 0 (characteristic equation)
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* - tr(A) = Σλᵢ (trace equals sum of eigenvalues)
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* - det(A) = Πλᵢ (determinant equals product of eigenvalues)
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* - Aⁿv = λⁿv (matrix power property)
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*/
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/**************************************************************************
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* EIGENVALUE AND EIGENVECTOR AXIOMS
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**************************************************************************/
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/**
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* @test Axiom: Av = λv (basic eigenvalue equation)
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* For each eigenvalue λ and corresponding eigenvector v, the equation Av = λv must hold
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*
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* @dataProvider dataProviderForEigenvalueAxioms
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* @param array $A
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* @throws \Exception
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*/
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public function testBasicEigenvalueEquation(array $A)
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{
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// Given
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$A = MatrixFactory::create($A);
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// When
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$eigenvalues = $A->eigenvalues();
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$eigenvectors = $A->eigenvectors();
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// Then
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for ($i = 0; $i < \count($eigenvalues); $i++) {
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$λ = $eigenvalues[$i];
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$v = new Vector($eigenvectors->getColumn($i));
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$Av = $A->vectorMultiply($v);
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$λv = $v->scalarMultiply($λ);
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// Av should equal λv
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$this->assertEqualsWithDelta($Av->getVector(), $λv->getVector(), 1e-6);
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}
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}
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/**
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* @test Axiom: det(A - λI) = 0 (characteristic equation)
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* For each eigenvalue λ, the determinant of (A - λI) should be zero
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*
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* @dataProvider dataProviderForEigenvalueAxioms
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* @param array $A
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* @throws \Exception
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*/
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public function testCharacteristicEquation(array $A)
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{
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// Given
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$A = MatrixFactory::create($A);
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$I = MatrixFactory::identity($A->getM());
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// When
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$eigenvalues = $A->eigenvalues();
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// Then
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foreach ($eigenvalues as $λ) {
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$λI = $I->scalarMultiply($λ);
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$A_minus_λI = $A->subtract($λI);
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$det = $A_minus_λI->det();
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$this->assertEqualsWithDelta(0, $det, 1e-6);
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}
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}
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/**
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* @test Axiom: tr(A) = Σλᵢ (trace equals sum of eigenvalues)
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* The trace of a matrix equals the sum of its eigenvalues
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*
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* @dataProvider dataProviderForEigenvalueAxioms
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* @param array $A
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* @throws \Exception
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*/
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public function testTraceEqualsSumOfEigenvalues(array $A)
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{
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// Given
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$A = MatrixFactory::create($A);
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// When
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$trace = $A->trace();
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$eigenvalues = $A->eigenvalues();
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$sum_of_eigenvalues = \array_sum($eigenvalues);
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// Then
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$this->assertEqualsWithDelta($trace, $sum_of_eigenvalues, 1e-6);
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}
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/**
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* @test Axiom: det(A) = Πλᵢ (determinant equals product of eigenvalues)
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* The determinant of a matrix equals the product of its eigenvalues
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*
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* @dataProvider dataProviderForEigenvalueAxioms
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* @param array $A
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* @throws \Exception
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*/
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public function testDeterminantEqualsProductOfEigenvalues(array $A)
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{
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// Given
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$A = MatrixFactory::create($A);
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// When
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$det = $A->det();
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$eigenvalues = $A->eigenvalues();
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$product_of_eigenvalues = \array_product($eigenvalues);
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// Then
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$this->assertEqualsWithDelta($det, $product_of_eigenvalues, 1e-6);
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}
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/**
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* @test Axiom: Aⁿv = λⁿv (matrix power property)
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* For eigenvalue λ and eigenvector v, raising the matrix to power n gives A^n v = λ^n v
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*
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* @dataProvider dataProviderForEigenvalueMatrixPower
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* @param array $A
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* @param int $n
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* @throws \Exception
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*/
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public function testMatrixPowerProperty(array $A, int $n)
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{
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// Given
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$A = MatrixFactory::create($A);
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// When
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$eigenvalues = $A->eigenvalues();
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$eigenvectors = $A->eigenvectors();
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// Calculate Aⁿ by repeated multiplication
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$An = MatrixFactory::identity($A->getM());
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for ($i = 0; $i < $n; $i++) {
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$An = $An->multiply($A);
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}
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// Then
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for ($i = 0; $i < \count($eigenvalues); $i++) {
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$λ = $eigenvalues[$i];
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$v = new Vector($eigenvectors->getColumn($i));
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$Aⁿv = $An->vectorMultiply($v);
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$λⁿv = $v->scalarMultiply(pow($λ, $n));
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// Aⁿv should equal λⁿv
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for ($j = 0; $j < $Aⁿv->getN(); $j++) {
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$this->assertEqualsWithDelta($λⁿv->get($j), $Aⁿv->get($j), 1e-5);
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}
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}
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}
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/**************************************************************************
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* DATA PROVIDERS FOR EIGENVALUE TESTS
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**************************************************************************/
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/**
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* Data provider for eigenvalue axiom tests with well-conditioned matrices
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* @return array
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*/
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public function dataProviderForEigenvalueAxioms(): array
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{
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return [
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// 2x2 diagonal matrix (simple eigenvalues)
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[
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[
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[2, 0],
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[0, 3],
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],
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],
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// 2x2 symmetric matrix
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[
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[
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[1, 2],
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[2, 1],
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],
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],
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// 2x2 general matrix
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[
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[
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[0, 1],
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[-2, -3],
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],
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],
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// 3x3 diagonal matrix
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[
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[
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[1, 0, 0],
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[0, 2, 0],
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[0, 0, 3],
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],
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],
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// 3x3 symmetric matrix
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[
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[
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[2, -1, 0],
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[-1, 2, -1],
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[0, -1, 2],
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],
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],
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// 3x3 upper triangular matrix
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[
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[
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[1, 2, 3],
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[0, 4, 5],
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[0, 0, 6],
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],
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],
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// 3x3 general matrix
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[
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[
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[-2, -4, 2],
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[-2, 1, 2],
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[4, 2, 5],
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],
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],
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];
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}
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/**
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* Data provider for matrix power eigenvalue tests
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* @return array
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*/
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public function dataProviderForEigenvalueMatrixPower(): array
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{
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return [
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// 2x2 diagonal matrix with power 2
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[
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[
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[2, 0],
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[0, 3],
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],
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2,
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],
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// 2x2 diagonal matrix with power 3
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[
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[
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[2, 0],
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[0, 3],
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],
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3,
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],
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// 2x2 symmetric matrix with power 2
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[
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[
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[1, 2],
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[2, 1],
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],
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2,
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],
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// 3x3 diagonal matrix with power 2
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[
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[
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[1, 0, 0],
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[0, 2, 0],
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[0, 0, 3],
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],
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2,
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],
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// 3x3 upper triangular matrix with power 2
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[
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[
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[1, 1, 0],
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[0, 2, 1],
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[0, 0, 3],
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],
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2,
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],
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];
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}
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}

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