﻿ 矩阵广义迹与映射之间的关系研究

# 矩阵广义迹与映射之间的关系研究Research on the Relationship between Generalized Trace of Matrix and Mapping

Abstract: The relationship between generalized trace of matrix and mapping is studied. The basic operations of the matrix are combined. The relationship between trace of matrix and mapping is extended to that of generalized trace of general matrix, generalized trace of Hadamarad product and Kronecker product of special matrix and mapping.

1. 引言

2. 预备知识

$C={\left({c}_{ij}\right)}_{ms×nt}=A{\otimes }_{L}B=\left[\begin{array}{cccc}{a}_{11}B& {a}_{12}B& \cdots & {a}_{1n}B\\ {a}_{21}B& {a}_{22}B& \cdots & {a}_{2n}B\\ ⋮& ⋮& \ddots & ⋮\\ {a}_{m1}B& {a}_{m2}B& \cdots & {a}_{mn}B\end{array}\right]$

$C={\left({c}_{ij}\right)}_{m×n}$ 为A与B的左Kronecker积，记作： $C=A{\otimes }_{L}B$

$D={\left({d}_{ij}\right)}_{ms×nt}={A}_{R}\otimes B=\left[\begin{array}{cccc}{b}_{11}A& {b}_{12}A& \cdots & {b}_{1t}A\\ {b}_{21}A& {b}_{22}A& \cdots & {b}_{2t}A\\ ⋮& ⋮& \ddots & ⋮\\ {b}_{s1}A& {b}_{s2}A& \cdots & {b}_{st}A\end{array}\right]$

$D={\left({d}_{ij}\right)}_{ms×nt}$ 为A与B的右Kronecker积，记作： $D={A}_{R}\otimes B$

$f\left(\alpha +\beta \right)=f\left(\alpha \right)+f\left(\beta \right)$

$f\left(k\alpha \right)=kf\left(\alpha \right)$

3. 主要结论

1) $\forall A,B\in {M}_{n}\left(F\right)$，有 $f\left(A+B\right)=f\left(A\right)+f\left(B\right)$

2) $\forall A\in {M}_{n×n}\left(F\right)$，数 $k\in F$，有 $f\left(kA\right)=kf\left(A\right)$

3) $\forall A,B\in {M}_{n}\left(F\right)$，有 $f\left(AB\right)=f\left(BA\right)$

4) $f\left({E}_{n}\right)=n$

$f\left(A\right)=trA$$\forall A\in {M}_{n}\left(F\right)$ 都成立。

$f\left({E}_{n}\right)=f\left({E}_{11}+{E}_{22}+\cdots +{E}_{nn}\right)=f\left({E}_{11}\right)+f\left({E}_{22}\right)+\cdots +f\left({E}_{nn}\right)=n$

$f\left(A\right)=f\left(\underset{j=1}{\overset{n}{\sum }}\underset{i=1}{\overset{n}{\sum }}{a}_{ij}{E}_{ij}\right)=\underset{j=1}{\overset{n}{\sum }}\underset{i=1}{\overset{n}{\sum }}{a}_{ij}f\left({E}_{ij}\right)=\underset{i=1}{\overset{n}{\sum }}{a}_{ii}=trA$

1) $\forall A,B\in {M}_{m×n}\left(F\right)$，有 $f\left(A+B\right)=f\left(A\right)+f\left(B\right)$

2) $\forall A\in {M}_{m×n}\left(F\right)$，数 $k\in F$，有 $f\left(kA\right)=kf\left(A\right)$

3) $\forall A\in {M}_{m×n}\left(F\right)$$\forall B\in {M}_{n×m}\left(F\right)$，有 $f\left(AB\right)=f\left(BA\right)$

4) $f\left(\left[\begin{array}{cc}{E}_{\mathrm{min}\left(m,n\right)}& {O}_{\mathrm{min}\left(m,n\right)×\left[n-\mathrm{min}\left(m,n\right)\right]}\\ {O}_{\left[m-\mathrm{min}\left(m,n\right)\right]×\mathrm{min}\left(m,n\right)}& {O}_{\left[m-\mathrm{min}\left(m,n\right)\right]×\left[n-\mathrm{min}\left(m,n\right)\right]}\end{array}\right]\right)=\mathrm{min}\left(m,n\right)$

$f\left(A\right)=trA$$\forall A\in {M}_{m×n}\left(F\right)$ 都成立。

$m=n$ 时，

$\begin{array}{l}f\left(\left[\begin{array}{cc}{E}_{\mathrm{min}\left(m,n\right)}& {O}_{\mathrm{min}\left(m,n\right)×\left[n-\mathrm{min}\left(m,n\right)\right]}\\ {O}_{\left[m-\mathrm{min}\left(m,n\right)\right]×\mathrm{min}\left(m,n\right)}& {O}_{\left[m-\mathrm{min}\left(m,n\right)\right]×\left[n-\mathrm{min}\left(m,n\right)\right]}\end{array}\right]\right)\\ =f\left({E}_{m}\right)=f\left({E}_{11}+{E}_{22}+\cdots +{E}_{mm}\right)=f\left({E}_{11}\right)+f\left({E}_{22}\right)+\cdots +f\left({E}_{mm}\right)=m.\end{array}$

$m 时，

$\begin{array}{l}f\left(\left[\begin{array}{cc}{E}_{\mathrm{min}\left(m,n\right)}& {O}_{\mathrm{min}\left(m,n\right)×\left[n-\mathrm{min}\left(m,n\right)\right]}\\ {O}_{\left[m-\mathrm{min}\left(m,n\right)\right]×\mathrm{min}\left(m,n\right)}& {O}_{\left[m-\mathrm{min}\left(m,n\right)\right]×\left[n-\mathrm{min}\left(m,n\right)\right]}\end{array}\right]\right)\\ =f\left(\left[{E}_{m},{O}_{m×\left(n-m\right)}\right]\right)=f\left(\left[{E}_{11},{O}_{m×\left(n-m\right)}\right]+\left[{E}_{22},{O}_{m×\left(n-m\right)}\right]+\cdots +\left[{E}_{mm},{O}_{m×\left(n-m\right)}\right]\right)\\ =f\left(\left[{E}_{11},{O}_{m×\left(n-m\right)}\right]\right)+f\left(\left[{E}_{22},{O}_{m×\left(n-m\right)}\right]\right)+\cdots +f\left(\left[{E}_{mm},{O}_{m×\left(n-m\right)}\right]\right)=m.\end{array}$

$m>n$ 时，

$\begin{array}{l}f\left(\left[\begin{array}{cc}{E}_{\mathrm{min}\left(m,n\right)}& {O}_{\mathrm{min}\left(m,n\right)×\left[n-\mathrm{min}\left(m,n\right)\right]}\\ {O}_{\left[m-\mathrm{min}\left(m,n\right)\right]×\mathrm{min}\left(m,n\right)}& {O}_{\left[m-\mathrm{min}\left(m,n\right)\right]×\left[n-\mathrm{min}\left(m,n\right)\right]}\end{array}\right]\right)\\ =f\left(\left[\begin{array}{c}{E}_{n}\\ {O}_{\left(m-n\right)×n}\end{array}\right]\right)=f\left(\left[\begin{array}{c}{E}_{11}\\ {O}_{\left(m-n\right)×n}\end{array}\right]+\left[\begin{array}{c}{E}_{22}\\ {O}_{\left(m-n\right)×n}\end{array}\right]+\cdots +\left[\begin{array}{c}{E}_{nn}\\ {O}_{\left(m-n\right)×n}\end{array}\right]\right)\\ =f\left(\left[\begin{array}{c}{E}_{11}\\ {O}_{\left(m-n\right)×n}\end{array}\right]\right)+f\left(\left[\begin{array}{c}{E}_{22}\\ {O}_{\left(m-n\right)×n}\end{array}\right]\right)+\cdots +f\left(\left[\begin{array}{c}{E}_{nn}\\ {O}_{\left(m-n\right)×n}\end{array}\right]\right)=n.\end{array}$

$m=n$ 时，

$f\left(A\right)=f\left(\underset{j=1}{\overset{m}{\sum }}\underset{i=1}{\overset{m}{\sum }}{a}_{ij}{E}_{ij}\right)=\underset{j=1}{\overset{m}{\sum }}\underset{i=1}{\overset{m}{\sum }}{a}_{ij}f\left({E}_{ij}\right)=\underset{i=1}{\overset{m}{\sum }}{a}_{ii}=trA$.

$m 时，

$\begin{array}{c}f\left(A\right)=f\left(\underset{j=1}{\overset{m}{\sum }}\underset{i=1}{\overset{m}{\sum }}{a}_{ij}{E}_{ij}+\underset{j=m+1}{\overset{n}{\sum }}\underset{i=1}{\overset{m}{\sum }}{a}_{ij}{O}_{ij}\right)\\ =\underset{j=1}{\overset{n}{\sum }}\underset{i=1}{\overset{n}{\sum }}{a}_{ij}f\left({E}_{ij}\right)+\underset{j=1}{\overset{n}{\sum }}\underset{i=n+1}{\overset{m}{\sum }}{a}_{ij}f\left({O}_{ij}\right)=\underset{i=1}{\overset{m}{\sum }}{a}_{ii}+0=trA;\end{array}$

$m>n$ 时，

$\begin{array}{c}f\left(A\right)=f\left(\underset{j=1}{\overset{n}{\sum }}\underset{i=1}{\overset{n}{\sum }}{a}_{ij}{E}_{ij}+\underset{j=1}{\overset{n}{\sum }}\underset{i=n+1}{\overset{m}{\sum }}{a}_{ij}{O}_{ij}\right)\\ =\underset{j=1}{\overset{n}{\sum }}\underset{i=1}{\overset{n}{\sum }}{a}_{ij}f\left({E}_{ij}\right)+\underset{j=1}{\overset{n}{\sum }}\underset{i=n+1}{\overset{m}{\sum }}{a}_{ij}f\left({O}_{ij}\right)=\underset{i=1}{\overset{n}{\sum }}{a}_{ii}+0=trA.\end{array}$

1) $\forall A,B,C\in {M}_{m×n}\left(F\right)$，有 $f\left(A\circ C+B\circ C\right)=f\left(A\circ C\right)+f\left(B\circ C\right)$

2) $\forall A,B\in {M}_{m×n}\left(F\right)$，数 $k\in F$，有 $f\left(kA\circ B\right)=kf\left(A\circ B\right)$

3) $\forall A,B\in {M}_{m×n}\left(F\right)$$\forall D\in {M}_{n×m}\left(F\right)$，有 $f\left[\left(A\circ B\right)D\right]=f\left[D\left(A\circ B\right)\right]$

4) $f\left(A\circ \left[\begin{array}{cc}{E}_{\mathrm{min}\left(m,n\right)}& {O}_{\mathrm{min}\left(m,n\right)×\left[n-\mathrm{min}\left(m,n\right)\right]}\\ {O}_{\left[m-\mathrm{min}\left(m,n\right)\right]×\mathrm{min}\left(m,n\right)}& {O}_{\left[m-\mathrm{min}\left(m,n\right)\right]×\left[n-\mathrm{min}\left(m,n\right)\right]}\end{array}\right]\right)=\underset{i=1}{\overset{\mathrm{min}\left(m,n\right)}{\sum }}{a}_{ii}$

$f\left(A\circ B\right)=tr\left(A\circ B\right)$$\forall A,B\in {M}_{m×n}\left(F\right)$ 都成立。

$m=n$ 时，

$\begin{array}{l}f\left(A\circ \left[\begin{array}{cc}{E}_{\mathrm{min}\left(m,n\right)}& {O}_{\mathrm{min}\left(m,n\right)×\left[n-\mathrm{min}\left(m,n\right)\right]}\\ {O}_{\left[m-\mathrm{min}\left(m,n\right)\right]×\mathrm{min}\left(m,n\right)}& {O}_{\left[m-\mathrm{min}\left(m,n\right)\right]×\left[n-\mathrm{min}\left(m,n\right)\right]}\end{array}\right]\right)\\ =f\left(A\circ {E}_{m}\right)=f\left(A\circ {E}_{11}+A\circ {E}_{22}+\cdots +A\circ {E}_{mm}\right)\\ =f\left(A\circ {E}_{11}\right)+f\left(A\circ {E}_{22}\right)+\cdots +f\left(A\circ {E}_{mm}\right)=\underset{i=1}{\overset{m}{\sum }}{a}_{ii}.\end{array}$

$m 时，

$\begin{array}{l}f\left(A\circ \left[\begin{array}{cc}{E}_{\mathrm{min}\left(m,n\right)}& {O}_{\mathrm{min}\left(m,n\right)×\left[n-\mathrm{min}\left(m,n\right)\right]}\\ {O}_{\left[m-\mathrm{min}\left(m,n\right)\right]×\mathrm{min}\left(m,n\right)}& {O}_{\left[m-\mathrm{min}\left(m,n\right)\right]×\left[n-\mathrm{min}\left(m,n\right)\right]}\end{array}\right]\right)\\ =f\left(A\circ \left[{E}_{m},{O}_{m×\left(n-m\right)}\right]\right)=f\left(A\circ \left[{E}_{11},{O}_{m×\left(n-m\right)}\right]+A\circ \left[{E}_{22},{O}_{m×\left(n-m\right)}\right]+\cdots +A\circ \left[{E}_{mm},{O}_{m×\left(n-m\right)}\right]\right)\\ =f\left(A\circ \left[{E}_{11},{O}_{m×\left(n-m\right)}\right]\right)+f\left(A\circ \left[{E}_{22},{O}_{m×\left(n-m\right)}\right]\right)+\cdots +f\left(A\circ \left[{E}_{mm},{O}_{m×\left(n-m\right)}\right]\right)=\underset{i=1}{\overset{m}{\sum }}{a}_{ii}.\end{array}$

$m>n$ 时，

$\begin{array}{l}f\left(A\circ \left[\begin{array}{cc}{E}_{\mathrm{min}\left(m,n\right)}& {O}_{\mathrm{min}\left(m,n\right)×\left[n-\mathrm{min}\left(m,n\right)\right]}\\ {O}_{\left[m-\mathrm{min}\left(m,n\right)\right]×\mathrm{min}\left(m,n\right)}& {O}_{\left[m-\mathrm{min}\left(m,n\right)\right]×\left[n-\mathrm{min}\left(m,n\right)\right]}\end{array}\right]\right)\\ =f\left(A\circ \left[\begin{array}{c}{E}_{n}\\ {O}_{\left(m-n\right)×n}\end{array}\right]\right)=f\left(A\circ \left[\begin{array}{c}{E}_{11}\\ {O}_{\left(m-n\right)×n}\end{array}\right]+A\circ \left[\begin{array}{c}{E}_{22}\\ {O}_{\left(m-n\right)×n}\end{array}\right]+\cdots +A\circ \left[\begin{array}{c}{E}_{nn}\\ {O}_{\left(m-n\right)×n}\end{array}\right]\right)\\ =f\left(A\circ \left[\begin{array}{c}{E}_{11}\\ {O}_{\left(m-n\right)×n}\end{array}\right]\right)+f\left(A\circ \left[\begin{array}{c}{E}_{22}\\ {O}_{\left(m-n\right)×n}\end{array}\right]\right)+\cdots +f\left(A\circ \left[\begin{array}{c}{E}_{nn}\\ {O}_{\left(m-n\right)×n}\end{array}\right]\right)=\underset{i=1}{\overset{n}{\sum }}{a}_{ii}.\end{array}$

$m=n$ 时，

$\begin{array}{c}f\left(A\circ B\right)=f\left(\underset{j=1}{\overset{m}{\sum }}\underset{i=1}{\overset{m}{\sum }}{a}_{ij}{b}_{ij}{E}_{ij}\right)=\underset{j=1}{\overset{m}{\sum }}\underset{i=1}{\overset{m}{\sum }}{a}_{ij}{b}_{ij}f\left({E}_{ij}\right)\\ =\underset{i=1}{\overset{m}{\sum }}{a}_{ii}{b}_{ii}=tr\left(A\circ B\right).\end{array}$

$m 时，

$\begin{array}{c}f\left(A\circ B\right)=f\left(\underset{j=1}{\overset{m}{\sum }}\underset{i=1}{\overset{m}{\sum }}{a}_{ij}{b}_{ij}{E}_{ij}+\underset{j=m+1}{\overset{n}{\sum }}\underset{i=1}{\overset{m}{\sum }}{a}_{ij}{b}_{ij}{O}_{ij}\right)\\ =\underset{j=1}{\overset{n}{\sum }}\underset{i=1}{\overset{n}{\sum }}{a}_{ij}{b}_{ij}f\left({E}_{ij}\right)+\underset{j=1}{\overset{n}{\sum }}\underset{i=n+1}{\overset{m}{\sum }}{a}_{ij}{b}_{ij}f\left({O}_{ij}\right)\\ =\underset{i=1}{\overset{m}{\sum }}{a}_{ii}{b}_{ii}+0=tr\left(A\circ B\right);\end{array}$

$m>n$ 时，

$\begin{array}{c}f\left(A\circ B\right)=f\left(\underset{j=1}{\overset{n}{\sum }}\underset{i=1}{\overset{n}{\sum }}{a}_{ij}{b}_{ij}{E}_{ij}+\underset{j=1}{\overset{n}{\sum }}\underset{i=n+1}{\overset{m}{\sum }}{a}_{ij}{b}_{ij}{O}_{ij}\right)\\ =\underset{j=1}{\overset{n}{\sum }}\underset{i=1}{\overset{n}{\sum }}{a}_{ij}{b}_{ij}f\left({E}_{ij}\right)+\underset{j=1}{\overset{n}{\sum }}\underset{i=n+1}{\overset{m}{\sum }}{a}_{ij}{b}_{ij}f\left({O}_{ij}\right)\\ =\underset{i=1}{\overset{n}{\sum }}{a}_{ii}{b}_{ii}+0=tr\left(A\circ B\right).\end{array}$

1) $\forall A\in {M}_{m×n}\left(F\right)$$B,C\in {M}_{s×t}\left(F\right)$，有 $f\left(A{\otimes }_{R}B+A{\otimes }_{R}C\right)=f\left(A{\otimes }_{R}B\right)+f\left(A{\otimes }_{R}B\right)$

2) $\forall A\in {M}_{m×n}\left(F\right)$$B\in {M}_{s×t}\left(F\right)$，数 $k\in F$，有 $f\left(kA{\otimes }_{L}B\right)=kf\left(A{\otimes }_{L}B\right)$

3) $\forall A\in {M}_{m×n}\left(F\right)$$B\in {M}_{s×t}\left(F\right)$$\forall D\in {M}_{nt×ms}\left(F\right)$，有 $f\left[\left(A{\otimes }_{L}B\right)D\right]=f\left[D\left(A{\otimes }_{L}B\right)\right]$

4) $f\left(\left(A{\otimes }_{L}B\right)\left[\begin{array}{cc}{E}_{\mathrm{min}\left(ms,nt\right)}& {O}_{\mathrm{min}\left(ms,nt\right)×\left[nt-\mathrm{min}\left(ms,nt\right)\right]}\\ {O}_{\left[ms-\mathrm{min}\left(ms,nt\right)\right]×\mathrm{min}\left(ms,nt\right)}& {O}_{\left[ms-\mathrm{min}\left(ms,nt\right)\right]×\left[nt-\mathrm{min}\left(ms,nt\right)\right]}\end{array}\right]\right)=\underset{i=1}{\overset{\mathrm{min}\left(m,n\right)}{\sum }}\underset{j=1}{\overset{\mathrm{min}\left(s,t\right)}{\sum }}{b}_{jj}{a}_{ii}$

$f\left(A{\otimes }_{L}B\right)=tr\left(A{\otimes }_{L}B\right)$$\forall A\in {M}_{m×n}\left(F\right)$$B\in {M}_{s×t}\left(F\right)$ 都成立。

1) $\forall A\in {M}_{m×n}\left(F\right)$$B,C\in {M}_{s×t}\left(F\right)$，有 $f\left(A{}_{R}\otimes \text{\hspace{0.17em}}B+A{}_{R}\otimes \text{\hspace{0.17em}}C\right)=f\left(A{}_{R}\otimes \text{\hspace{0.17em}}B\right)+f\left(A{}_{R}\otimes \text{\hspace{0.17em}}C\right)$

2) $\forall A\in {M}_{m×n}\left(F\right)$$B\in {M}_{s×t}\left(F\right)$，数 $k\in F$，有 $f\left(kA{}_{R}\otimes \text{\hspace{0.17em}}B\right)=kf\left(A{}_{R}\otimes \text{\hspace{0.17em}}B\right)$

3) $\forall A\in {M}_{m×n}\left(F\right)$$B\in {M}_{s×t}\left(F\right)$$\forall D\in {M}_{nt×ms}\left(F\right)$，有 $f\left[\left(A{}_{R}\otimes \text{\hspace{0.17em}}B\right)D\right]=f\left[D\left(A{}_{R}\otimes \text{\hspace{0.17em}}B\right)\right]$

4) $f\left(\left(A{}_{R}\otimes B\right)\left[\begin{array}{cc}{E}_{\mathrm{min}\left(ms,nt\right)}& {O}_{\mathrm{min}\left(ms,nt\right)×\left[nt-\mathrm{min}\left(ms,nt\right)\right]}\\ {O}_{\left[ms-\mathrm{min}\left(ms,nt\right)\right]×\mathrm{min}\left(ms,nt\right)}& {O}_{\left[ms-\mathrm{min}\left(ms,nt\right)\right]×\left[nt-\mathrm{min}\left(ms,nt\right)\right]}\end{array}\right]\right)=\underset{j=1}{\overset{\mathrm{min}\left(s,t\right)}{\sum }}\underset{i=1}{\overset{\mathrm{min}\left(m,n\right)}{\sum }}{a}_{ii}{b}_{jj}$

$f\left(A{}_{R}\otimes \text{\hspace{0.17em}}B\right)=tr\left(A{}_{R}\otimes \text{\hspace{0.17em}}B\right)$$\forall A\in {M}_{m×n}\left(F\right)$$B\in {M}_{s×t}\left(F\right)$ 都成立。

4. 小结

NOTES

*通讯作者。

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[2] 刘兴祥, 李姣, 朱磊, 等. 矩阵的两种特殊运算的广义迹及拉伸运算的关系[J]. 河南科学, 2014, 32(1): 7-11.

[3] 张禾瑞, 郝鈵新. 高等代数[M]. 北京: 高等教育出版社, 2007.

[4] 辛轶. 矩阵广义迹[J]. 宁德师专学报(自然科学版), 2007, 19(1): 4-6.

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