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Kronecker Product

Tool to calculate a Kronecker matrix product in computer algebra. The Kronecker product is a special case of tensor multiplication on matrices.

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Tag(s) : Mathematics,Algebra,Symbolic Computation

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# Kronecker Product

## Kronecker Product

Tool to calculate a Kronecker matrix product in computer algebra. The Kronecker product is a special case of tensor multiplication on matrices.

### How to multiply 2 matrices with Kronecker?

Consider $$M_1=[a_{ij}]$$ a matrix of $$m$$ lines and $$n$$ columns and $$M_2=[b_{ij}]$$ a matrix of $$p$$ lines and $$q$$ columns. The Kronecker product is noted with $$M_1 \otimes M_2 = [c_{ij}]$$ is a larger matrix of $$m \times p$$ lines and $$n \times q$$ columns, with : $$\forall i, j : c_{ij} = a_{ij}.B$$

$$M=\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} \otimes \begin{bmatrix} 7 & 8 \\ 9 & 10 \end{bmatrix} = \begin{bmatrix} 7 & 8 & 14 & 16 & 21 & 24 \\ 9 & 10 & 18 & 20 & 27 & 30 \\ 28 & 32 & 35 & 40 & 42 & 48 \\ 36 & 40 & 45 & 50 & 54 & 60 \end{bmatrix}$$

This product is not equivalent to the classical multiplication" target="_blank">matrix product, $$M_1 \otimes M_2 \neq M_1 \dot M_2$$

### What are matrix Kronecker multiplication properties?

The Kronecker product suport associativity :

$$A \otimes (B+ \lambda\ \cdot C) = (A \otimes B) + \lambda (A \otimes C) \\ (A + \lambda\ \cdot B) \otimes C = (A \otimes C) + \lambda (B \otimes C) \\ A \otimes ( B \otimes C) = (A \otimes B) \otimes C \\ (A \otimes B) (C \otimes D) = (A C) \otimes (B D)$$

But is is non-commutative

$$A \otimes B \neq B \otimes A$$

It has also some properties:

- Distributivity over matrix transpose: $$( A \otimes B )^T = A^T \otimes B^T$$

- Distributivity over matrix traces: $$\operatorname{Tr}( A \otimes B ) = \operatorname{Tr}( A ) \operatorname{Tr}( B )$$

- Distributivity over matrix determinants: $$\operatorname{det}( A \otimes B ) = \operatorname{det}( A )^{m} \operatorname{det}( B )^{n}$$