linalg
::kroneckerProduct
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 (m p)
×(n q) 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.
Note that the Kronecker product is only implemented for matrices
over the domains Dom::Matrix
, Dom::SquareMatrix
or Dom::MatrixGroup
.
In particular, this includes matrices created by matrix
.
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.
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:
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:

Matrices of the domains 
Matrix of the same type as A
or B
.