Kronecker product

Definition. Let $A=(a_{ij})$ be a $n\times n$ matrix and let $B$ be a $m\times m$ matrix. Then the Kronecker product of $A$ and $B$ is the $mn\times mn$ block matrix

$A\otimes B$ $=$

The Kronecker product is also known as the direct product or the tensor product [1].

Fundamental properties [1, 2]

1. The product is bilinear. If $k$ is a scalar, and $A,B$ and $C$ are square matrices, such that $B$ and $C$ are of the same order, then

   $A\otimes (B+C)$ $=$ $A\otimes B+A\otimes C,$

   $(B+C)\otimes A$ $=$ $B\otimes A+C\otimes A,$

   $k(A\otimes B)$ $=$ $(kA)\otimes B=A\otimes (kB).$

2. If $A,B,C,D$ are square matrices such that the products $AC$ and $BD$ exist, then $(A\otimes B)(C\otimes D)$ exists and

   $(A\otimes B)(C\otimes D)$ $=$ $AC\otimes BD.$

   If $A$ and $B$ are invertible matrices, then

   ${(A\otimes B)}^{-1}$ $=$ $A^{-1}\otimes B^{-1}.$

3. If $A$ and $B$ are square matrices, then for the transpose ($A^T$) we have

   ${(A\otimes B)}^T$ $=$ $A^T\otimes B^T.$

4. Let $A$ and $B$ be square matrices of orders $n$ and $m$, respectively. If $\{{\lambda }_i|i=1,\mathrm{\dots },n\}$ are the eigenvalues of $A$ and $\{{\mu }_j|j=1,\mathrm{\dots },m\}$ are the eigenvalues of $B$, then $\{{\lambda }_i{\mu }_j|i=1,\mathrm{\dots },n,j=1,\mathrm{\dots },m\}$ are the eigenvalues of $A\otimes B$. Also,

   $det(A\otimes B)$ $=$ ${(detA)}^m{(detB)}^n,$

   $rank(A\otimes B)$ $=$ $rankArankB,$

   $trace(A\otimes B)$ $=$ $traceAtraceB,$

Title	Kronecker product
Canonical name	KroneckerProduct
Date of creation	2013-03-22 13:33:31
Last modified on	2013-03-22 13:33:31
Owner	Mathprof (13753)
Last modified by	Mathprof (13753)
Numerical id	7
Author	Mathprof (13753)
Entry type	Definition
Classification	msc 15-00
Synonym	tensor product (for matrices)
Synonym	direct product