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

作者: 武真真 :东明县职业中等专业学校,山东 菏泽; 刘兴祥 :延安大学数学与计算机科学学院,陕西 延安;

关键词: 矩阵的广义迹Hadamard积Kronecker积映射Matrix Generalized Trace Hadamard Product Kronecker Product Mapping

摘要: 研究了矩阵广义迹与映射之间的关系。结合矩阵的基本运算,将方阵的迹与映射之间的关系,推广到一般矩阵和两个特殊矩阵(Hadamard积、Kronecker积)的广义迹与映射之间的关系。

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

矩阵运算之间的关系是矩阵研究中的一个重要内容,而矩阵的迹运算在定义推广之后的研究显得十分重要。本文结合矩阵的基本运算,在方阵的迹与映射关系的基础上探究了一般矩阵的广义迹与映射之间的关系,并讨论了特殊矩阵Hadamard积和Kronecker积的广义迹与映射之间的关系。

2. 预备知识

定义1 [1] 设 A = ( a i j ) m × n ,则将 i = 1 min ( m , n ) a i i 称为矩阵A的广义迹,记作 t r ( A )

定义2 [2] 设 A = ( a i j ) m × n B = ( b i j ) m × n ,称 C = ( c i j ) m × n 为A与B的Hadamard积,其中 c i j = a i j × b i j ,记作 C = A B

定义3 [2] 设 A = ( a i j ) m × n B = ( b i j ) s × t ,有

C = ( c i j ) m s × n t = A L B = [ a 11 B a 12 B a 1 n B a 21 B a 22 B a 2 n B a m 1 B a m 2 B a m n B ]

C = ( c i j ) m × n 为A与B的左Kronecker积,记作: C = A L B

定义4 [2] 设 A = ( a i j ) m × n B = ( b i j ) s × t ,有

D = ( d i j ) m s × n t = A R B = [ b 11 A b 12 A b 1 t A b 21 A b 22 A b 2 t A b s 1 A b s 2 A b s t A ]

D = ( d i j ) m s × n t 为A与B的右Kronecker积,记作: D = A R B

定义5 [3] 设 V 1 , V 2 是数域F上的线性空间,若对 V 1 中任意两个向量 α , β 和任意的 k F ,都有

f ( α + β ) = f ( α ) + f ( β )

f ( k α ) = k f ( α )

则称f是 V 1 V 2 的一个映射。

3. 主要结论

引理 [4] 如果定义 f : M n ( F ) F 是一个映射,且满足以下的条件:

1) A , B M n ( F ) ,有 f ( A + B ) = f ( A ) + f ( B )

2) A M n × n ( F ) ,数 k F ,有 f ( k A ) = k f ( A )

3) A , B M n ( F ) ,有 f ( A B ) = f ( B A )

4) f ( E n ) = n

f ( A ) = t r A A M n ( F ) 都成立。

证明:设 E i j 为n阶的基础矩阵,由条件(3)知 f ( E i i ) = f ( E i j E j i ) = f ( E j i E i j ) = f ( E j j ) ,所以 f ( E i i ) = 1 .

另一方面,当 i j E i j = E i k E k j ,则 f ( E i k E k j ) = f ( E k i E j k ) = f ( 0 ) = 0

又由条件(1)和条件(4)知:

f ( E n ) = f ( E 11 + E 22 + + E n n ) = f ( E 11 ) + f ( E 22 ) + + f ( E n n ) = n

由上述可知,如果设 A = ( a i j ) n × n ,由条件(2)可得

f ( A ) = f ( j = 1 n i = 1 n a i j E i j ) = j = 1 n i = 1 n a i j f ( E i j ) = i = 1 n a i i = t r A

综上可证: f ( A ) = t r A A M n ( F ) 都成立。

定理1如果定义 f : M m × n ( F ) F 是一个映射,且满足以下的条件:

1) A , B M m × n ( F ) ,有 f ( A + B ) = f ( A ) + f ( B )

2) A M m × n ( F ) ,数 k F ,有 f ( k A ) = k f ( A )

3) A M m × n ( F ) B M n × m ( F ) ,有 f ( A B ) = f ( B A )

4) f ( [ E min ( m , n ) O min ( m , n ) × [ n min ( m , n ) ] O [ m min ( m , n ) ] × min ( m , n ) O [ m min ( m , n ) ] × [ n min ( m , n ) ] ] ) = min ( m , n )

f ( A ) = t r A A M m × n ( F ) 都成立。

证明:设 E i j min ( m , n ) 阶的基础矩阵, O i j min ( m , n ) × [ n min ( m , n ) ] 阶或 [ m min ( m , n ) ] × min ( m , n ) 阶零矩阵,由条件(3)知: f ( E i i ) = f ( E i j E j i ) = f ( E j i E i j ) = f ( E j j ) ,所以 f ( E i i ) = 1

另一方面,当 i j E i j = E i k E k j ,则 f ( E i k E k j ) = f ( E k i E j k ) = f ( 0 ) = 0

又由条件(1)和条件(4)知:

m = n 时,

f ( [ E min ( m , n ) O min ( m , n ) × [ n min ( m , n ) ] O [ m min ( m , n ) ] × min ( m , n ) O [ m min ( m , n ) ] × [ n min ( m , n ) ] ] ) = f ( E m ) = f ( E 11 + E 22 + + E m m ) = f ( E 11 ) + f ( E 22 ) + + f ( E m m ) = m .

m < n 时,

f ( [ E min ( m , n ) O min ( m , n ) × [ n min ( m , n ) ] O [ m min ( m , n ) ] × min ( m , n ) O [ m min ( m , n ) ] × [ n min ( m , n ) ] ] ) = f ( [ E m , O m × ( n m ) ] ) = f ( [ E 11 , O m × ( n m ) ] + [ E 22 , O m × ( n m ) ] + + [ E m m , O m × ( n m ) ] ) = f ( [ E 11 , O m × ( n m ) ] ) + f ( [ E 22 , O m × ( n m ) ] ) + + f ( [ E m m , O m × ( n m ) ] ) = m .

m > n 时,

f ( [ E min ( m , n ) O min ( m , n ) × [ n min ( m , n ) ] O [ m min ( m , n ) ] × min ( m , n ) O [ m min ( m , n ) ] × [ n min ( m , n ) ] ] ) = f ( [ E n O ( m n ) × n ] ) = f ( [ E 11 O ( m n ) × n ] + [ E 22 O ( m n ) × n ] + + [ E n n O ( m n ) × n ] ) = f ( [ E 11 O ( m n ) × n ] ) + f ( [ E 22 O ( m n ) × n ] ) + + f ( [ E n n O ( m n ) × n ] ) = n .

综上可得: f ( [ E min ( m , n ) O min ( m , n ) × [ n min ( m , n ) ] O [ m min ( m , n ) ] × min ( m , n ) O [ m min ( m , n ) ] × [ n min ( m , n ) ] ] ) = min ( m , n )

由上述可知,如果设 A = ( a i j ) m × n ,由条件(2)可得

m = n 时,

f ( A ) = f ( j = 1 m i = 1 m a i j E i j ) = j = 1 m i = 1 m a i j f ( E i j ) = i = 1 m a i i = t r A .

m < n 时,

f ( A ) = f ( j = 1 m i = 1 m a i j E i j + j = m + 1 n i = 1 m a i j O i j ) = j = 1 n i = 1 n a i j f ( E i j ) + j = 1 n i = n + 1 m a i j f ( O i j ) = i = 1 m a i i + 0 = t r A ;

m > n 时,

f ( A ) = f ( j = 1 n i = 1 n a i j E i j + j = 1 n i = n + 1 m a i j O i j ) = j = 1 n i = 1 n a i j f ( E i j ) + j = 1 n i = n + 1 m a i j f ( O i j ) = i = 1 n a i i + 0 = t r A .

综上可证: f ( A ) = t r A A M m × n ( F ) 都成立。

定理2如果定义 f : M m × n ( F ) F 是一个映射,且满足以下的条件:

1) A , B , C M m × n ( F ) ,有 f ( A C + B C ) = f ( A C ) + f ( B C )

2) A , B M m × n ( F ) ,数 k F ,有 f ( k A B ) = k f ( A B )

3) A , B M m × n ( F ) D M n × m ( F ) ,有 f [ ( A B ) D ] = f [ D ( A B ) ]

4) f ( A [ E min ( m , n ) O min ( m , n ) × [ n min ( m , n ) ] O [ m min ( m , n ) ] × min ( m , n ) O [ m min ( m , n ) ] × [ n min ( m , n ) ] ] ) = i = 1 min ( m , n ) a i i

f ( A B ) = t r ( A B ) A , B M m × n ( F ) 都成立。

证明:设 E i j min ( m , n ) 阶的基础矩阵, O i j min ( m , n ) × [ n min ( m , n ) ] 阶或 [ m min ( m , n ) ] × min ( m , n ) 阶零矩阵,由条件(3)知: f ( E i i ) = f ( E i j E j i ) = f ( E j i E i j ) = f ( E j j ) ,所以 f ( E i i ) = 1

另一方面,当 i j E i j = E i k E k j ,则 f ( E i k E k j ) = f ( E k i E j k ) = f ( 0 ) = 0

又由条件(1)和条件(4)知:

m = n 时,

f ( A [ E min ( m , n ) O min ( m , n ) × [ n min ( m , n ) ] O [ m min ( m , n ) ] × min ( m , n ) O [ m min ( m , n ) ] × [ n min ( m , n ) ] ] ) = f ( A E m ) = f ( A E 11 + A E 22 + + A E m m ) = f ( A E 11 ) + f ( A E 22 ) + + f ( A E m m ) = i = 1 m a i i .

m < n 时,

f ( A [ E min ( m , n ) O min ( m , n ) × [ n min ( m , n ) ] O [ m min ( m , n ) ] × min ( m , n ) O [ m min ( m , n ) ] × [ n min ( m , n ) ] ] ) = f ( A [ E m , O m × ( n m ) ] ) = f ( A [ E 11 , O m × ( n m ) ] + A [ E 22 , O m × ( n m ) ] + + A [ E m m , O m × ( n m ) ] ) = f ( A [ E 11 , O m × ( n m ) ] ) + f ( A [ E 22 , O m × ( n m ) ] ) + + f ( A [ E m m , O m × ( n m ) ] ) = i = 1 m a i i .

m > n 时,

f ( A [ E min ( m , n ) O min ( m , n ) × [ n min ( m , n ) ] O [ m min ( m , n ) ] × min ( m , n ) O [ m min ( m , n ) ] × [ n min ( m , n ) ] ] ) = f ( A [ E n O ( m n ) × n ] ) = f ( A [ E 11 O ( m n ) × n ] + A [ E 22 O ( m n ) × n ] + + A [ E n n O ( m n ) × n ] ) = f ( A [ E 11 O ( m n ) × n ] ) + f ( A [ E 22 O ( m n ) × n ] ) + + f ( A [ E n n O ( m n ) × n ] ) = i = 1 n a i i .

综上可得: f ( A [ E min ( m , n ) O min ( m , n ) × [ n min ( m , n ) ] O [ m min ( m , n ) ] × min ( m , n ) O [ m min ( m , n ) ] × [ n min ( m , n ) ] ] ) = i = 1 min ( m , n ) j = 1 min ( s , t ) b j j a i i

由上述可知,如果设 A = ( a i j ) m × n B = ( b i j ) m × n ,由条件(2)可得

m = n 时,

f ( A B ) = f ( j = 1 m i = 1 m a i j b i j E i j ) = j = 1 m i = 1 m a i j b i j f ( E i j ) = i = 1 m a i i b i i = t r ( A B ) .

m < n 时,

f ( A B ) = f ( j = 1 m i = 1 m a i j b i j E i j + j = m + 1 n i = 1 m a i j b i j O i j ) = j = 1 n i = 1 n a i j b i j f ( E i j ) + j = 1 n i = n + 1 m a i j b i j f ( O i j ) = i = 1 m a i i b i i + 0 = t r ( A B ) ;

m > n 时,

f ( A B ) = f ( j = 1 n i = 1 n a i j b i j E i j + j = 1 n i = n + 1 m a i j b i j O i j ) = j = 1 n i = 1 n a i j b i j f ( E i j ) + j = 1 n i = n + 1 m a i j b i j f ( O i j ) = i = 1 n a i i b i i + 0 = t r ( A B ) .

综上可证: f ( A B ) = t r ( A B ) A , B M m × n ( F ) 都成立。

定理3如果定义 f : M m × n ( F ) F 是一个映射,且满足以下的条件:

1) A M m × n ( F ) B , C M s × t ( F ) ,有 f ( A R B + A R C ) = f ( A R B ) + f ( A R B )

2) A M m × n ( F ) B M s × t ( F ) ,数 k F ,有 f ( k A L B ) = k f ( A L B )

3) A M m × n ( F ) B M s × t ( F ) D M n t × m s ( F ) ,有 f [ ( A L B ) D ] = f [ D ( A L B ) ]

4) f ( ( A L B ) [ E min ( m s , n t ) O min ( m s , n t ) × [ n t min ( m s , n t ) ] O [ m s min ( m s , n t ) ] × min ( m s , n t ) O [ m s min ( m s , n t ) ] × [ n t min ( m s , n t ) ] ] ) = i = 1 min ( m , n ) j = 1 min ( s , t ) b j j a i i

f ( A L B ) = t r ( A L B ) A M m × n ( F ) B M s × t ( F ) 都成立。

证明过程可以类比定理1和定理2推出,由于过程较为繁琐,所以在此省略。

推论如果定义 f : M m × n ( F ) F 是一个映射,且满足以下的条件:

1) A M m × n ( F ) B , C M s × t ( F ) ,有 f ( A R B + A R C ) = f ( A R B ) + f ( A R C )

2) A M m × n ( F ) B M s × t ( F ) ,数 k F ,有 f ( k A R B ) = k f ( A R B )

3) A M m × n ( F ) B M s × t ( F ) D M n t × m s ( F ) ,有 f [ ( A R B ) D ] = f [ D ( A R B ) ]

4) f ( ( A R B ) [ E min ( m s , n t ) O min ( m s , n t ) × [ n t min ( m s , n t ) ] O [ m s min ( m s , n t ) ] × min ( m s , n t ) O [ m s min ( m s , n t ) ] × [ n t min ( m s , n t ) ] ] ) = j = 1 min ( s , t ) i = 1 min ( m , n ) a i i b j j

f ( A R B ) = t r ( A R B ) A M m × n ( F ) B M s × t ( F ) 都成立。

4. 小结

通过研究将矩阵广义迹与映射这两种概念联系起来,说明很多概念之间是有联系的。在这篇文章中主要结合矩阵的基本运算,在方阵与映射关系的基础上,研究了一般矩阵的广义迹和特殊矩阵Hadamard积以及Kronecker积的广义迹与映射之间的关系。矩阵的广义迹可能与其它内容也是有联系的,有待于以后继续去发现。

基金项目

国家自然科学基金项目(12161086)。

参考文献

NOTES

*通讯作者。

文章引用: 武真真 , 刘兴祥 (2021) 矩阵广义迹与映射之间的关系研究。 应用数学进展, 10, 3337-3342. doi: 10.12677/AAM.2021.1010350

参考文献

[1] 杨楠, 刘兴祥, 岳育英. m × n矩阵k次广义迹[J]. 河南科学, 2012, 30(2): 149-152.

[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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