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kron

Kronecker tensor product

Description

example

K = kron(A,B) returns the Kronecker tensor product of matrices A and B. If A is an m-by-n matrix and B is a p-by-q matrix, then kron(A,B) is an m*p-by-n*q matrix formed by taking all possible products between the elements of A and the matrix B.

Examples

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Create a block diagonal matrix.

Create a 4-by-4 identity matrix and a 2-by-2 matrix that you want to be repeated along the diagonal.

A = eye(4);
B = [1 -1;-1 1];

Use kron to find the Kronecker tensor product.

K = kron(A,B)
K = 8×8

1    -1     0     0     0     0     0     0
-1     1     0     0     0     0     0     0
0     0     1    -1     0     0     0     0
0     0    -1     1     0     0     0     0
0     0     0     0     1    -1     0     0
0     0     0     0    -1     1     0     0
0     0     0     0     0     0     1    -1
0     0     0     0     0     0    -1     1

The result is an 8-by-8 block diagonal matrix.

Expand the size of a matrix by repeating elements.

Create a 2-by-2 matrix of ones and a 2-by-3 matrix whose elements you want to repeat.

A = [1 2 3; 4 5 6];
B = ones(2);

Calculate the Kronecker tensor product using kron.

K = kron(A,B)
K = 4×6

1     1     2     2     3     3
1     1     2     2     3     3
4     4     5     5     6     6
4     4     5     5     6     6

The result is a 4-by-6 block matrix.

This example visualizes a sparse Laplacian operator matrix.

The matrix representation of the discrete Laplacian operator on a two-dimensional, n-by- n grid is a n*n-by- n*n sparse matrix. There are at most five nonzero elements in each row or column. You can generate the matrix as the Kronecker product of one-dimensional difference operators. In this example n = 5.

n = 5;
I = speye(n,n);
E = sparse(2:n,1:n-1,1,n,n);
D = E+E'-2*I;
A = kron(D,I)+kron(I,D);

Visualize the sparsity pattern with spy.

spy(A,'k') Input Arguments

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Input matrices, specified as scalars, vectors, or matrices. If either A or B is sparse, then kron multiplies only nonzero elements and the result is also sparse.

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | logical
Complex Number Support: Yes

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Kronecker Tensor Product

If A is an m-by-n matrix and B is a p-by-q matrix, then the Kronecker tensor product of A and B is a large matrix formed by multiplying B by each element of A

$A\otimes B=\left[\begin{array}{ccc}\begin{array}{cc}{a}_{11}B& {a}_{12}B\end{array}& \cdots & {a}_{1n}B\\ \begin{array}{cc}\begin{array}{c}{a}_{21}B\\ ⋮\end{array}& \begin{array}{c}{a}_{22}B\\ ⋮\end{array}\end{array}& \begin{array}{c}\cdots \\ \ddots \end{array}& \begin{array}{c}{a}_{2n}B\\ ⋮\end{array}\\ \begin{array}{cc}{a}_{m1}B& {a}_{m2}B\end{array}& \cdots & {a}_{mn}B\end{array}\right].$

For example, two simple 2-by-2 matrices produce

$\begin{array}{l}A=\left[\begin{array}{cc}1& -2\\ -1& 0\end{array}\right],\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}B=\left[\begin{array}{cc}4& -3\\ 2& 3\end{array}\right]\\ \\ A\otimes B=\left[\begin{array}{cccc}1\text{\hspace{0.17em}}·\text{\hspace{0.17em}}4& 1\text{\hspace{0.17em}}·\text{\hspace{0.17em}}-3& -2\text{\hspace{0.17em}}·\text{\hspace{0.17em}}4& -2\text{\hspace{0.17em}}·\text{\hspace{0.17em}}-3\\ 1\text{\hspace{0.17em}}·\text{\hspace{0.17em}}2& 1\text{\hspace{0.17em}}·\text{\hspace{0.17em}}3& -2\text{\hspace{0.17em}}·\text{\hspace{0.17em}}2& -2\text{\hspace{0.17em}}·\text{\hspace{0.17em}}3\\ -1\text{\hspace{0.17em}}·\text{\hspace{0.17em}}4& -1\text{\hspace{0.17em}}·\text{\hspace{0.17em}}-3& 0\text{\hspace{0.17em}}·\text{\hspace{0.17em}}4& 0\text{\hspace{0.17em}}·\text{\hspace{0.17em}}-3\\ -1\text{\hspace{0.17em}}·\text{\hspace{0.17em}}2& -1\text{\hspace{0.17em}}·\text{\hspace{0.17em}}3& 0\text{\hspace{0.17em}}·\text{\hspace{0.17em}}2& 0\text{\hspace{0.17em}}·\text{\hspace{0.17em}}3\end{array}\right]=\left[\begin{array}{cccc}4& -3& -8& 6\\ 2& 3& -4& -6\\ -4& 3& 0& 0\\ -2& -3& 0& 0\end{array}\right].\end{array}$