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Kronecker product of matrices

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linalg::kroneckerProduct(A, <B, …>)


linalg::kroneckerProduct(A, B) computes the Kronecker product of two matrices A and B.

The Kronecker product (direct matrix product) of an m×n matrix A and a p×q matrix B is the (mp) ×(nq) matrix given in block form as


Componentwise: with I = p (i - 1) + k, J = q (j - 1) + l.

If A and B are matrices of the same matrix domain with the same component ring, the result is a matrix of the same type as A and B. If the domains or the component rings of A and B differ, linalg::kroneckerProduct tries to convert B into the domain type of A by A::dom::coerce. If this fails, conversion of A to the domain type of B is attempted. If no conversion is possible, an error is raised.

A call with more than two arguments produces linalg::kroneckerProduct(A, B, C) = linalg::kroneckerProduct(linalg::kroneckerProduct(A, B), C) etc.

A call with only one argument is possible. It returns the input matrix.


Example 1

We consider two matrices A and B with symbolic components:

A:= matrix([[a11, a12], [a21, a22]]);
B:= matrix([[b11, b12, b13], [b21, b22, b23]]);

The Kronecker product of A and B is computed by multiplying the matrix B with each of the components of the matrix A. The resulting block matrix is returned as a matrix of larger dimension:

linalg::kroneckerProduct(A, B);

delete A, B:

Example 2

An n×n matrix H with components in {- 1, 1} is called a Hadamard matrix if H multiplied with its transpose equals n times the n×n identity matrix. The matrix H defined below is a Hadamard matrix:

H:= matrix([[1, 1], [1, -1]]);
H * linalg::transpose(H) = 2 * matrix::identity(2);

Hadamard matrices play a role in the field of error correcting codes. A basic property of this type of matrices is that the Kronecker product of two Hadamard matrices is again a Hadamard matrix. We verify this statement for the matrix H:

H2:= linalg::kroneckerProduct(H, H);

Indeed, the matrix H2 is a again a Hadamard matrix:

H2 * linalg::transpose(H2) = 4 * matrix::identity(4);

delete H, H2:


A, B, …

Matrices of the domains Dom::Matrix, Dom::SquareMatrix or Dom::MatrixGroup

Return Values

Matrix of the same type as A or B.

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