矩陣廣義逆新的擾動上界

孔祥強

(菏澤學院數(shù)學系,山東菏澤 274000)

矩陣廣義逆新的擾動上界

孔祥強

(菏澤學院數(shù)學系,山東菏澤 274000)

利用矩陣的奇異值分解方法,研究了矩陣廣義逆的擾動上界,得到了在F-范數(shù)下矩陣廣義逆的擾動上界定理,所得定理推廣并徹底改進了近期的相關(guān)結(jié)果.相應的數(shù)值算例驗證了定理的有效性.

廣義逆；奇異值分解；Frobenius范數(shù)；擾動

1 引言

對矩陣廣義逆擾動上界的研究,普遍采用的是B?-A?的分解形式,其中B為A擾動后的矩陣.而本文采用了矩陣奇異值分解的方法,得到了優(yōu)于近期相關(guān)結(jié)論的定理,通過數(shù)值算例進一步說明了定理的正確性.

2 幾個定義和定理

表1 μ的取值

文獻[5]研究了與矩陣廣義逆相關(guān)的秩等式和不等式,文獻[6]中對矩陣廣義逆的擾動界作了更加詳細、系統(tǒng)的研究,但都沒能改進定理2.1的擾動上界.下面給出的結(jié)論改進了定理2.1,并得出了相應的推論.

3 主要結(jié)果

4 數(shù)值算例

[1]Chen Y L.Generalized Inverses:Theory and Methods[M].Nanjing:Nanjing Normal University Press,2005.

[2]孫繼廣.矩陣擾動分析[M].北京:科學出版社,2001.

[3]徐樹方.矩陣計算的理論與方法[M].北京:北京大學出版社,1995.

[4]Wedit P A.Perturbation theory for pseudo-inverses[J].BIT,1973,13:217-232.

[6]Stewart G W.On the perturbation of the pseudo-inverse,projections,and linear squares problems[J].SIAM J.Rev.,1977,19(4):634-662.

[7]Chen X S,Li W.A note on tne perturbation bounds of eigenspaces for Hermitian matrices[J].J.Comput. Appl.Math.,2006,196(1):338-346.

New perturbation bounds of generalized inverse

Kong Xiangqiang

(Department of Mathematics,Heze University,Heze274000,China)

Using the singular value decomposition method,the paper studies the perturbation bounds of the generalized inverse.In this paper,some theorems for perturbation bounds of the generalized inverse are obtained according to the Frobenius norm,some related results are improved.Advantages of results obtained are illustrated by a numerical example.

generalized inverse,singular value decomposition,Frobenius norm,perturbation

O241.6

A

1008-5513(2012)04-0516-07

2012-02-22.

山東省統(tǒng)計局重點課題項目(KT11048)；山東省教育科學“十二五”規(guī)劃重點課題項目(2011GG049).

孔祥強(1983-),碩士,助教,研究方向：應用數(shù)學.

2010 MSC:15A09