Kronecker Product

Question 1

How to multiply 2 matrices with Kronecker?

Answer

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 $

Question 2

What are matrix Kronecker multiplication properties?

Answer

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} $

Question 3

Why is this multiplication called Kronecker?

Answer

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

Kronecker Product