| 任意域上解约束矩阵方程的Cramer法则 |
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| 引用本文： | 张显,王国荣.任意域上解约束矩阵方程的Cramer法则[J].黑龙江大学自然科学学报,2006,23(5):637-641. |
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| 作者姓名： | 张显 王国荣 |
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| 作者单位： | 1. 黑龙江大学,数学科学学院,黑龙江,哈尔滨,150080
2. 上海师范大学,数学科学学院,上海,200234 |
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| 基金项目： | 黑龙江省教委海外学者基金 |
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| 摘 要： | Fm×n表示域F上所有m×n矩阵的集合.R(A)和Nr(A)分别表示矩阵A∈Fm×n的列空间和核空间.若m=n,用Ind(A)定义矩阵A的指标.给出了求一类约束矩阵方程WAWXWBW=D,R(X)R((AW)k1),Nr(X)Nr((WB)k2)的唯一解的Cramer法则,其中A∈Fm×n,W∈Fn×m,B∈Fp×q,W∈Fq×p,D∈Fn×p,R(D)R((WA)k2),Nr(D)Nr((BWk1),k1=Ind(AW),k2=Ind(WA),k1=Ind(BW),k2=Ind(WB).这将15-17]中的结果从复数域推广到任意域.
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| 关 键 词： | 域 Cramer法则 W-Drazin逆 约束矩阵方程 指标 |
| 文章编号： | 1001-7011(2006)05-0637-05 |
| 修稿时间： | 2006年3月15日 |
Cramer rules for solving restricted matrix equations over arbitrary fields |
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| ZHANG Xian,WANG Guo-rong.Cramer rules for solving restricted matrix equations over arbitrary fields[J].Journal of Natural Science of Heilongjiang University,2006,23(5):637-641. |
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| Authors: | ZHANG Xian WANG Guo-rong |
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| Abstract: | Let Fm×n be the set of all m × n matrices over a field F. For a matrix A ∈ Fm×n, we denote the column space and right kernel space of A by R(A) and Nr(A) , respectively. If m = n, we also denote by Ind(A) the index of A. A Cramer rule for finding the unique solution of a class of restricted matrix equationsWAWX (～W)B (～W) = D , R( X) ()R( ( AW) k1 ) , Nr( X) () Nr( ( (～W)B )(~k)2 )is presented, whereA ∈Fm×n , W∈Fn×m, B∈Fp×q, W∈Fq×p, D∈Fn×p , R(D)()R((WA)k2), Nr(D)()Nr( (B (～W))(~k)1 ), k1= Ind(AW), k2= Ind( WA), (~k)1 = Ind( B (～W)) and (~k)2 = Ind((～W)B). This generalizes those results in 15,16,17 ] from the field of complex numbers to any field. |
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| Keywords: | field Cramer rule W-weighted Drazin inverse restricted matrix equation index |
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