Given an matrix and a matrix , their Kronecker product , also called their matrix direct product, is an matrix with elements defined by
(1)
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where
(2)
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(3)
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For example, the matrix direct product of the matrix and the matrix is given by the following matrix,
(4)
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(5)
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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)
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and
(7)
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are given by and . Then
(8)
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is determined by
(9)
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