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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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Kronecker Product -

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# Kronecker Product

## Kronecker Product

### How to multiply 2 matrices with Kronecker?

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

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

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

### What are matrix Kronecker multiplication properties?

The Kronecker product supports 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 Kronecker product is non-commutative

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

Kronecker product has also some distributivity 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: $\operatorname{det}( A \otimes B ) = \operatorname{det}( A )^{m} \operatorname{det}( B )^{n}$

### Why is this multiplication called Kronecker?

The name is a tribute to the German mathematician Leopold Kronecker.

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