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Kronecker Product


Given an m×n matrix A and a p×q matrix B, their Kronecker product C=A tensor B, also called their matrix direct product, is an (mp)×(nq) matrix with elements defined by

 c_(alphabeta)=a_(ij)b_(kl),
(1)

where

alpha=p(i-1)+k
(2)
beta=q(j-1)+l.
(3)

For example, the matrix direct product of the 2×2 matrix A and the 3×2 matrix B is given by the following 6×4 matrix,

A tensor B=[a_(11)B a_(12)B; a_(21)B a_(22)B]
(4)
=[a_(11)b_(11) a_(11)b_(12) a_(12)b_(11) a_(12)b_(12); a_(11)b_(21) a_(11)b_(22) a_(12)b_(21) a_(12)b_(22); a_(11)b_(31) a_(11)b_(32) a_(12)b_(31) a_(12)b_(32); a_(21)b_(11) a_(21)b_(12) a_(22)b_(11) a_(22)b_(12); a_(21)b_(21) a_(21)b_(22) a_(22)b_(21) a_(22)b_(22); a_(21)b_(31) a_(21)b_(32) a_(22)b_(31) a_(22)b_(32)].
(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

 S:V_1->W_1
(6)

and

 T:V_2->W_2
(7)

are given by S(x)=Ax and T(y)=By. Then

 S tensor T:V_1 tensor V_2->W_1 tensor W_2
(8)

is determined by

 S tensor T(x tensor y)=(Ax) tensor (By)=(A tensor B)(x tensor y).
(9)

See also

Direct Product, Graph Tensor Product, Matrix Multiplication, Tensor Direct Product

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References

Schafer, R. D. An Introduction to Nonassociative Algebras. New York: Dover, p. 12, 1996.

Referenced on Wolfram|Alpha

Kronecker Product

Cite this as:

Weisstein, Eric W. "Kronecker Product." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/KroneckerProduct.html

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