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 following 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