kron

Kronecker tensor product

Syntax

K = kron(A,B)

Description

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

Block Diagonal Matrix

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.

Repeat Matrix Elements

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.

Sparse Laplacian Operator 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

A,B — Input matrices

scalars | vectors | matrices

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

More About

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

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

Extended Capabilities

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

GPU Code Generation
Generate CUDA® code for NVIDIA® GPUs using GPU Coder™.

Thread-Based Environment

Run code in the background using MATLAB® backgroundPool or accelerate code with Parallel Computing Toolbox™ ThreadPool.

This function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

The kron function fully supports GPU arrays. To run the function on a GPU, specify the input data as a gpuArray (Parallel Computing Toolbox). For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Distributed Arrays

Partition large arrays across the combined memory of your cluster using Parallel Computing Toolbox™.

This function fully supports distributed arrays. For more information, see Run MATLAB Functions with Distributed Arrays (Parallel Computing Toolbox).

Version History

Introduced before R2006a