# 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

 $\displaystyle A\otimes B$ $\displaystyle=$ $\displaystyle\left(\begin{array}[]{ccc}a_{11}B&\cdots&a_{1n}B\\ \vdots&\ddots&\vdots\\ a_{n1}B&\cdots&a_{nn}B\\ \end{array}\right).$

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

Fundamental properties [1, 2]

1. 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

 $\displaystyle A\otimes(B+C)$ $\displaystyle=$ $\displaystyle A\otimes B+A\otimes C,$ $\displaystyle(B+C)\otimes A$ $\displaystyle=$ $\displaystyle B\otimes A+C\otimes A,$ $\displaystyle k(A\otimes B)$ $\displaystyle=$ $\displaystyle(kA)\otimes B=A\otimes(kB).$
2. 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

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

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

 $\displaystyle(A\otimes B)^{-1}$ $\displaystyle=$ $\displaystyle A^{-1}\otimes B^{-1}.$
3. 3.

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

 $\displaystyle(A\otimes B)^{T}$ $\displaystyle=$ $\displaystyle A^{T}\otimes B^{T}.$
4. 4.

Let $A$ and $B$ be square matrices of orders $n$ and $m$, respectively. If $\{\lambda_{i}|i=1,\ldots,n\}$ are the eigenvalues of $A$ and $\{\mu_{j}|j=1,\ldots,m\}$ are the eigenvalues of $B$, then $\{\lambda_{i}\mu_{j}|i=1,\ldots,n,\,j=1,\ldots,m\}$ are the eigenvalues of $A\otimes B$. Also,

 $\displaystyle\det(A\otimes B)$ $\displaystyle=$ $\displaystyle(\det A)^{m}(\det B)^{n},$ $\displaystyle\mathop{\mathrm{rank}}(A\otimes B)$ $\displaystyle=$ $\displaystyle\mathop{\mathrm{rank}}A\,\mathop{\mathrm{rank}}B,$ $\displaystyle\mathop{\mathrm{trace}}(A\otimes B)$ $\displaystyle=$ $\displaystyle\mathop{\mathrm{trace}}A\,\mathop{\mathrm{trace}}B,$

## References

• 1 H. Eves, Elementary Matrix Theory, Dover publications, 1980.
• 2 T. Kailath, A.H. Sayed, B. Hassibi, Linear estimation, Prentice Hall, 2000
Title Kronecker product KroneckerProduct 2013-03-22 13:33:31 2013-03-22 13:33:31 Mathprof (13753) Mathprof (13753) 7 Mathprof (13753) Definition msc 15-00 tensor product (for matrices) direct product