Kronecker Product

Given an matrix and a matrix , their Kronecker product , also called their matrix direct product, is an matrix with elements defined by

(1)

where

			(2)
			(3)

For example, the matrix direct product of the matrix and the matrix is given by the following matrix,

			(4)
			(5)

The matrix direct product is implemented in the Wolfram Language as KroneckerProduct[a, b].

The matrix direct product gives the matrix of the linear transformation induced by the vector space tensor product of the original vector spaces. More precisely, suppose that

(6)

and

(7)

are given by and . Then

(8)

is determined by

(9)