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有對稱半正定解的充要條件及解的表達(dá)式.
矩陣方程;對稱半正定解;廣義奇異值分解;廣義逆矩陣
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)對稱矩陣與實(shí)對稱半正定矩陣的集合.A≥0表示A是對稱半正定矩陣,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)的對稱解,文[3]研究了(1.1)的對稱正定解.本文將討論矩陣方程(1.1)的對稱半正定解,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(1.1)中的矩陣對作如下廣義奇異值分解(GSVD)[4]:
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[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.
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[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,湖南邵陽人,助教,碩士,主要從事矩陣?yán)碚撗芯?