kron (MATLAB Function Reference)

Kronecker tensor product

Syntax

K = kron(X,Y)

Description

K = kron(X,Y)

returns the Kronecker tensor product of X and Y. The result is a large array formed by taking all possible products between the elements of X and those of Y. If X is m-by-n and Y is p-by-q, then kron(X,Y) is m*p-by-n*q.

Examples

If X is 2-by-3, then kron(X,Y) is

[ X(1,1)*Y X(1,2)*Y X(1,3)*Y
  X(2,1)*Y X(2,2)*Y X(2,3)*Y ]

The matrix representation of the discrete Laplacian operator on a two-dimensional, n-by-n grid is a n^2-by-n^2 sparse matrix. There are at most five nonzero elements in each row or column. The matrix can be generated as the Kronecker product of one-dimensional difference operators with these statements:

 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);

Plotting this with the spy function for n = 5 yields:

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