一類矩陣方程的對(duì)稱半正定解

李迎春,劉志宏

(紅河學(xué)院數(shù)學(xué)學(xué)院,云南蒙自661100)

一類矩陣方程的對(duì)稱半正定解

李迎春,劉志宏

(紅河學(xué)院數(shù)學(xué)學(xué)院,云南蒙自661100)

利用廣義奇異值分解和廣義逆給出了矩陣方程AXAT+B YBT=C有對(duì)稱半正定解的充要條件及解的表達(dá)式.

矩陣方程;對(duì)稱半正定解;廣義奇異值分解;廣義逆矩陣

0 引言

本文用Rm×n表示全體m×n實(shí)矩陣的集合,ORn×n表示階n實(shí)正交矩陣的集合,SRn×n與SRn×n≥0分別表示n階實(shí)對(duì)稱矩陣與實(shí)對(duì)稱半正定矩陣的集合.A≥0表示A是對(duì)稱半正定矩陣,AT,A+分別表示矩陣A的轉(zhuǎn)置及M oore-Penrose廣義逆,I為單位矩陣.

關(guān)于矩陣方程

的研究工作已有很多.文[1][2]研究了矩陣方程(1.1)的對(duì)稱解,文[3]研究了(1.1)的對(duì)稱正定解.本文將討論矩陣方程(1.1)的對(duì)稱半正定解,給出方程有解的充要條件及解的表達(dá)式.為研究方便,先對(duì)(1.1)中的矩陣對(duì)作如下廣義奇異值分解(GSVD)[4]:

1 引理

[1]Chang XW,Wang J S.The symmetric solution of the matrix equationAY+YA=C,AXAT+B YBT=C and (AXAT,B YBT)= (C,D)[J].LinearAlgebra Appl,1993,179:171-189.

[2]Xu G P,WeiM S,ZhengD S.On solution of the matrix equation AXB+CYD=F[J].LinearAlgebra and itsApplications,1998, 279:93-109.

[3]劉向華.矩陣方程AXAT+B YBT=C的對(duì)稱正定解[J].數(shù)學(xué)理論與應(yīng)用,2002,22(1):79-82.

[4]CCPaige,MASaunders.Towards a generalized singular value decomposition[J].SI AM JournalofNumericalAnalysis,1981,18:398-405.

[5]屠伯塤.亞正定矩陣I[J].數(shù)學(xué)學(xué)報(bào),1990,33(4):461-471.

[責(zé)任編輯 張燦邦]

Symmetric Sem ipositive Defin ite Solutions toMatrix Equations

LI Ying-chun,L IU Zhi-hong
(College ofMathematics of Honghe Univensity,Mengzi 661100,China)

Bymaking use of the generalized singular value decomposition ofmatrices and the generalized inverse matrix,the necessary and sufficient conditionsof the matrix equation AXAT+BYBT=C having the symmetric semipositive solutions and its expression of solutions are derived.

matrix equation;symmetric semipositive solutions;generalized singular value decomposition;generalized inverse matrix

book=8,ebook=139

013

A

1008-9128(2010)04-0034-03

2010-05-28

紅河學(xué)院博碩科研啟動(dòng)項(xiàng)目(XSS08012);云南省教育廳科研基金項(xiàng)目(09C0206)

李迎春(1979-),女,湖南邵陽(yáng)人,助教,碩士,主要從事矩陣?yán)碚撗芯?