# 克罗内克积

## 定义

${\displaystyle A\otimes B={\begin{bmatrix}a_{11}B&\cdots &a_{1n}B\\\vdots &\ddots &\vdots \\a_{m1}B&\cdots &a_{mn}B\end{bmatrix)).}$

${\displaystyle A\otimes B={\begin{bmatrix}a_{11}b_{11}&a_{11}b_{12}&\cdots &a_{11}b_{1q}&\cdots &\cdots &a_{1n}b_{11}&a_{1n}b_{12}&\cdots &a_{1n}b_{1q}\\a_{11}b_{21}&a_{11}b_{22}&\cdots &a_{11}b_{2q}&\cdots &\cdots &a_{1n}b_{21}&a_{1n}b_{22}&\cdots &a_{1n}b_{2q}\\\vdots &\vdots &\ddots &\vdots &&&\vdots &\vdots &\ddots &\vdots \\a_{11}b_{p1}&a_{11}b_{p2}&\cdots &a_{11}b_{pq}&\cdots &\cdots &a_{1n}b_{p1}&a_{1n}b_{p2}&\cdots &a_{1n}b_{pq}\\\vdots &\vdots &&\vdots &\ddots &&\vdots &\vdots &&\vdots \\\vdots &\vdots &&\vdots &&\ddots &\vdots &\vdots &&\vdots \\a_{m1}b_{11}&a_{m1}b_{12}&\cdots &a_{m1}b_{1q}&\cdots &\cdots &a_{mn}b_{11}&a_{mn}b_{12}&\cdots &a_{mn}b_{1q}\\a_{m1}b_{21}&a_{m1}b_{22}&\cdots &a_{m1}b_{2q}&\cdots &\cdots &a_{mn}b_{21}&a_{mn}b_{22}&\cdots &a_{mn}b_{2q}\\\vdots &\vdots &\ddots &\vdots &&&\vdots &\vdots &\ddots &\vdots \\a_{m1}b_{p1}&a_{m1}b_{p2}&\cdots &a_{m1}b_{pq}&\cdots &\cdots &a_{mn}b_{p1}&a_{mn}b_{p2}&\cdots &a_{mn}b_{pq}\end{bmatrix)).}$

### 例子

${\displaystyle {\begin{bmatrix}1&2\\3&1\\\end{bmatrix))\otimes {\begin{bmatrix}0&3\\2&1\\\end{bmatrix))={\begin{bmatrix}1\cdot 0&1\cdot 3&2\cdot 0&2\cdot 3\\1\cdot 2&1\cdot 1&2\cdot 2&2\cdot 1\\3\cdot 0&3\cdot 3&1\cdot 0&1\cdot 3\\3\cdot 2&3\cdot 1&1\cdot 2&1\cdot 1\\\end{bmatrix))={\begin{bmatrix}0&3&0&6\\2&1&4&2\\0&9&0&3\\6&3&2&1\end{bmatrix))}$.

## 特性

### 双线性和结合律

${\displaystyle A\otimes (B+C)=A\otimes B+A\otimes C\qquad {\mbox{(if ))B{\mbox{ and ))C{\mbox{ have the same size))),}$
${\displaystyle (A+B)\otimes C=A\otimes C+B\otimes C\qquad {\mbox{(if ))A{\mbox{ and ))B{\mbox{ have the same size))),}$
${\displaystyle (kA)\otimes B=A\otimes (kB)=k(A\otimes B),}$
${\displaystyle (A\otimes B)\otimes C=A\otimes (B\otimes C),}$

ABBA是排列等价的，也就是说，存在排列矩阵PQ，使得

${\displaystyle A\otimes B=P\,(B\otimes A)\,Q.}$

### 混合乘积性质

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

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

### 克罗内克和

${\displaystyle \mathbf {A} \oplus \mathbf {B} =\mathbf {A} \otimes \mathbf {I} _{m}+\mathbf {I} _{n}\otimes \mathbf {B} .}$

### 谱

${\displaystyle \lambda _{i}\mu _{j},\qquad i=1,\ldots ,n,\,j=1,\ldots ,q.}$

${\displaystyle \operatorname {tr} (\mathbf {A} \otimes \mathbf {B} )=\operatorname {tr} \mathbf {A} \,\operatorname {tr} \mathbf {B} \quad {\mbox{and))\quad \det(\mathbf {A} \otimes \mathbf {B} )=(\det \mathbf {A} )^{q}(\det \mathbf {B} )^{n}.}$

### 奇异值

${\displaystyle \sigma _{\mathbf {A} ,i},\qquad i=1,\ldots ,r_{\mathbf {A} }.}$

${\displaystyle \sigma _{\mathbf {B} ,i},\qquad i=1,\ldots ,r_{\mathbf {B} }.}$

${\displaystyle \sigma _{\mathbf {A} ,i}\sigma _{\mathbf {B} ,j},\qquad i=1,\ldots ,r_{\mathbf {A} },\,j=1,\ldots ,r_{\mathbf {B} }.}$

${\displaystyle \operatorname {rank} (\mathbf {A} \otimes \mathbf {B} )=\operatorname {rank} \mathbf {A} \,\operatorname {rank} \mathbf {B} .}$

### 转置

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

## 矩阵方程

${\displaystyle (B^{T}\otimes A)\,\operatorname {vec} (X)=\operatorname {vec} (AXB)=\operatorname {vec} (C).}$

## 参考文献

1. ^ Pages 401–402 of Dummit, David S.; Foote, Richard M., Abstract Algebra 2, New York: John Wiley and Sons, Inc., 1999, ISBN 0-471-36857-1
2. ^ D. E. Knuth: "Pre-Fascicle 0a: Introduction to Combinatorial Algorithms"页面存档备份，存于互联网档案馆）, zeroth printing (revision 2), to appear as part of D.E. Knuth: The Art of Computer Programming Vol. 4A
• Horn, Roger A.; Johnson, Charles R., Topics in Matrix Analysis, Cambridge University Press, 1991, ISBN 0-521-46713-6.
• Jain, Anil K., Fundamentals of Digital Image Processing, Prentice Hall, 1989, ISBN 0-13-336165-9.