Mathematics

The Kronecker Tensor Product

The Kronecker product, kron(X,Y), of two matrices is the larger matrix formed from all possible products of the elements of X with those of Y. If X is m-by-n and Y is p-by-q, then kron(X,Y) is mp-by-nq. The elements are arranged in the order

[X(1,1)*Y  X(1,2)*Y  . . .  X(1,n)*Y
                     . . .
 X(m,1)*Y  X(m,2)*Y  . . .  X(m,n)*Y]

The Kronecker product is often used with matrices of zeros and ones to build up repeated copies of small matrices. For example, if X is the 2-by-2 matrix

X =
       1     2
       3     4

and I = eye(2,2) is the 2-by-2 identity matrix, then the two matrices

kron(X,I)

and

kron(I,X)

are

       1     0     2     0
       0     1     0     2
       3     0     4     0
       0     3     0     4

and

       1     2     0     0
       3     4     0     0
       0     0     1     2
       0     0     3     4

The Identity Matrix

Vector and Matrix Norms