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kron

Kronecker tensor product

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Syntax

K = kron(A,B)

Description

example

K = kron(A,B)returns theKronecker tensor product矩阵的AandB. IfAis anm-by-nmatrix andBis ap-by-qmatrix, thenkron(A,B)is anm*p-by-n*qmatrix formed by taking all possible products between the elements ofAand the matrixB.

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];

Usekronto find the Kronecker tensor product.

K = kron(A,B)
K =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.

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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 usingkron.

K = kron(A,B)
K =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.

Open Live Script

This example visualizes a sparse Laplacian operator matrix.

The matrix representation of the discrete Laplacian operator on a two-dimensional,n-by-ngrid is an*n-by-n*nsparse 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 examplen = 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 withspy.

spy(A,'k')

Input Arguments

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Input matrices, specified as scalars, vectors, or matrices. If eitherAorBis sparse, thenkronmultiplies 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

More About

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Kronecker Tensor Product

IfAis anm-by-nmatrix andBis ap-by-qmatrix, then the Kronecker tensor product ofAandBis a large matrix formed by multiplyingBby each element ofA

A ⊗ B = [ a 11 B a 12 B ⋯ a 1 n B a 21 B ⋮ a 22 B ⋮ ⋯ ⋱ a 2 n B ⋮ a m 1 B a m 2 B ⋯ a m n B ] .

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

A = [ 1 − 2 − 1 0 ] , B = [ 4 − 3 2 3 ] A ⊗ B = [ 1 · 4 1 · − 3 − 2 · 4 − 2 · − 3 1 · 2 1 · 3 − 2 · 2 − 2 · 3 − 1 · 4 − 1 · − 3 0 · 4 0 · − 3 − 1 · 2 − 1 · 3 0 · 2 0 · 3 ] = [ 4 − 3 − 8 6 2 3 − 4 − 6 − 4 3 0 0 − 2 − 3 0 0 ] .

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

See Also

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Introduced before R2006a

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