Vehicle-cpp/hkpc/software/sympy/matrices/expressions/tests/test_kronecker.py

from sympy.core.mod import Mod
from sympy.core.numbers import I
from sympy.core.symbol import symbols
from sympy.functions.elementary.integers import floor
from sympy.matrices.dense import (Matrix, eye)
from sympy.matrices import MatrixSymbol, Identity
from sympy.matrices.expressions import det, trace

from sympy.matrices.expressions.kronecker import (KroneckerProduct,
                                                  kronecker_product,
                                                  combine_kronecker)


mat1 = Matrix([[1, 2 * I], [1 + I, 3]])
mat2 = Matrix([[2 * I, 3], [4 * I, 2]])

i, j, k, n, m, o, p, x = symbols('i,j,k,n,m,o,p,x')
Z = MatrixSymbol('Z', n, n)
W = MatrixSymbol('W', m, m)
A = MatrixSymbol('A', n, m)
B = MatrixSymbol('B', n, m)
C = MatrixSymbol('C', m, k)


def test_KroneckerProduct():
    assert isinstance(KroneckerProduct(A, B), KroneckerProduct)
    assert KroneckerProduct(A, B).subs(A, C) == KroneckerProduct(C, B)
    assert KroneckerProduct(A, C).shape == (n*m, m*k)
    assert (KroneckerProduct(A, C) + KroneckerProduct(-A, C)).is_ZeroMatrix
    assert (KroneckerProduct(W, Z) * KroneckerProduct(W.I, Z.I)).is_Identity


def test_KroneckerProduct_identity():
    assert KroneckerProduct(Identity(m), Identity(n)) == Identity(m*n)
    assert KroneckerProduct(eye(2), eye(3)) == eye(6)


def test_KroneckerProduct_explicit():
    X = MatrixSymbol('X', 2, 2)
    Y = MatrixSymbol('Y', 2, 2)
    kp = KroneckerProduct(X, Y)
    assert kp.shape == (4, 4)
    assert kp.as_explicit() == Matrix(
        [
            [X[0, 0]*Y[0, 0], X[0, 0]*Y[0, 1], X[0, 1]*Y[0, 0], X[0, 1]*Y[0, 1]],
            [X[0, 0]*Y[1, 0], X[0, 0]*Y[1, 1], X[0, 1]*Y[1, 0], X[0, 1]*Y[1, 1]],
            [X[1, 0]*Y[0, 0], X[1, 0]*Y[0, 1], X[1, 1]*Y[0, 0], X[1, 1]*Y[0, 1]],
            [X[1, 0]*Y[1, 0], X[1, 0]*Y[1, 1], X[1, 1]*Y[1, 0], X[1, 1]*Y[1, 1]]
        ]
    )


def test_tensor_product_adjoint():
    assert KroneckerProduct(I*A, B).adjoint() == \
        -I*KroneckerProduct(A.adjoint(), B.adjoint())
    assert KroneckerProduct(mat1, mat2).adjoint() == \
        kronecker_product(mat1.adjoint(), mat2.adjoint())


def test_tensor_product_conjugate():
    assert KroneckerProduct(I*A, B).conjugate() == \
        -I*KroneckerProduct(A.conjugate(), B.conjugate())
    assert KroneckerProduct(mat1, mat2).conjugate() == \
        kronecker_product(mat1.conjugate(), mat2.conjugate())


def test_tensor_product_transpose():
    assert KroneckerProduct(I*A, B).transpose() == \
        I*KroneckerProduct(A.transpose(), B.transpose())
    assert KroneckerProduct(mat1, mat2).transpose() == \
        kronecker_product(mat1.transpose(), mat2.transpose())


def test_KroneckerProduct_is_associative():
    assert kronecker_product(A, kronecker_product(
        B, C)) == kronecker_product(kronecker_product(A, B), C)
    assert kronecker_product(A, kronecker_product(
        B, C)) == KroneckerProduct(A, B, C)


def test_KroneckerProduct_is_bilinear():
    assert kronecker_product(x*A, B) == x*kronecker_product(A, B)
    assert kronecker_product(A, x*B) == x*kronecker_product(A, B)


def test_KroneckerProduct_determinant():
    kp = kronecker_product(W, Z)
    assert det(kp) == det(W)**n * det(Z)**m


def test_KroneckerProduct_trace():
    kp = kronecker_product(W, Z)
    assert trace(kp) == trace(W)*trace(Z)


def test_KroneckerProduct_isnt_commutative():
    assert KroneckerProduct(A, B) != KroneckerProduct(B, A)
    assert KroneckerProduct(A, B).is_commutative is False


def test_KroneckerProduct_extracts_commutative_part():
    assert kronecker_product(x * A, 2 * B) == x * \
        2 * KroneckerProduct(A, B)


def test_KroneckerProduct_inverse():
    kp = kronecker_product(W, Z)
    assert kp.inverse() == kronecker_product(W.inverse(), Z.inverse())


def test_KroneckerProduct_combine_add():
    kp1 = kronecker_product(A, B)
    kp2 = kronecker_product(C, W)
    assert combine_kronecker(kp1*kp2) == kronecker_product(A*C, B*W)


def test_KroneckerProduct_combine_mul():
    X = MatrixSymbol('X', m, n)
    Y = MatrixSymbol('Y', m, n)
    kp1 = kronecker_product(A, X)
    kp2 = kronecker_product(B, Y)
    assert combine_kronecker(kp1+kp2) == kronecker_product(A+B, X+Y)


def test_KroneckerProduct_combine_pow():
    X = MatrixSymbol('X', n, n)
    Y = MatrixSymbol('Y', n, n)
    assert combine_kronecker(KroneckerProduct(
        X, Y)**x) == KroneckerProduct(X**x, Y**x)
    assert combine_kronecker(x * KroneckerProduct(X, Y)
                             ** 2) == x * KroneckerProduct(X**2, Y**2)
    assert combine_kronecker(
        x * (KroneckerProduct(X, Y)**2) * KroneckerProduct(A, B)) == x * KroneckerProduct(X**2 * A, Y**2 * B)
    # cannot simplify because of non-square arguments to kronecker product:
    assert combine_kronecker(KroneckerProduct(A, B.T) ** m) == KroneckerProduct(A, B.T) ** m


def test_KroneckerProduct_expand():
    X = MatrixSymbol('X', n, n)
    Y = MatrixSymbol('Y', n, n)

    assert KroneckerProduct(X + Y, Y + Z).expand(kroneckerproduct=True) == \
        KroneckerProduct(X, Y) + KroneckerProduct(X, Z) + \
        KroneckerProduct(Y, Y) + KroneckerProduct(Y, Z)

def test_KroneckerProduct_entry():
    A = MatrixSymbol('A', n, m)
    B = MatrixSymbol('B', o, p)

    assert KroneckerProduct(A, B)._entry(i, j) == A[Mod(floor(i/o), n), Mod(floor(j/p), m)]*B[Mod(i, o), Mod(j, p)]