Kronecker Product

Definition

Kronecker multiplication, denoted by $\otimes$ , is defined for any two matrices $A_{n \times m}$ and $B_{p \times q}$ as

A \otimes B = a_{11} B ⋮ a_{n 1} B \dots ⋱ \dots a_{1 m} B ⋮ a_{nm} B

Kronecker matrix multiplication is a mapping

R^{n \times m} \times R^{p \times q} \to R^{n p \times m q}

$∣ A \otimes B ∣ = ∣ A ∣^{m} ∣ B ∣^{n}$
$(a A) \otimes (b B) = ab (A \otimes B) = (ab A) \otimes B = A \otimes (ab B)$ : Scalar distributive & commutative
$(A + B) \otimes (C) = A \otimes C + B \otimes C$ : Distributive
$(A \otimes B)^{T} = A^{T} \otimes B^{T}$
$(A \otimes B) (C \otimes D) = A C \otimes B D$
If $A, B$ square matrices, then $tr(A\otimesB) = tr (A) tr (B)$