Kronecker Product

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: $$ \forall i, j: c_{ij} = a_{ij}.B $$

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} $

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