矩阵方程的约束Hermitian最小二乘解

王方圆,李 莹,许洲慧,查秀秀

(1.聊城大学数学科学学院,山东聊城 252000;2.聊城市技师学院基础教学部，山东聊城 252000)

王方圆1,李 莹1,许洲慧2,查秀秀1

(1.聊城大学数学科学学院,山东聊城 252000;2.聊城市技师学院基础教学部，山东聊城 252000)

矩阵函数；矩阵方程；秩；惯性指数；最小二乘解

1 预备知识

引理2[10]设A∈Cm×n，B∈Cm×k,C∈Cl×n。则:

由引理2可得到以下秩公式：

引理3[11]设A∈Cm×n,B∈Cm×k,C∈Cl×n。则有：

引理4[12]设A∈Cm×n,B∈Cm×k,C∈Cl×n。若R(AQ)=R(A),R((PA)*)=R(A*)。则:

b)[12]设A,B,C,D,P与Q使得矩阵表达式D-CP+AQ+B有意义。则有：

引理7[13]给定A∈Cm×n,B∈Cn×p与C∈Cm×p,设X∈Cn×n为变量矩阵,假设矩阵方程AXB=C相容，则下列各项等价。

a)矩阵方程AXB=C存在Hermitian解;

c)矩阵方程AYB=C与B*YA*=C*有公共解Y。

X=A+B(A+)*+FAV+V*FA,

其中V∈Cn×n为任意矩阵。

记

则

2 约束Hermitian最小二乘解

由文献[6]知A1XB1=C1的Hermitian最小二乘解的通解表达式为

(1)

(2)

式(2)是一个关于3个变量U1,U2与U3的线性Hermitian矩阵函数，将其表示为

(3)

(4)

对式(3)应用引理9可得：

则

(5)

(6)

(7)

证明在式(4)的条件下,对式(3)应用引理9得：

(8)

利用矩阵的初等变换可得：

(9)

(10)

利用引理2—引理6及引理9计算得：

(11)

r(A1)-r(B1)=r(M2)-2r(A1)-2r(B1),

(12)

2r(A1)-r(B1)=r(N1)-3r(A1)-r(B1),

(13)

r(A1)-2r(B1)=r(N2)-2r(A1)-3r(B1),

(14)

(15)

将式(9)—式(15)分别代入式(8)即得式(5)—式(7)。

利用以上秩与惯性指数极值与引理1即得如下结论。

定理2矩阵A1,B1,C1,A2,C2如定理1所述,记M1,M2,N1,N2,M,G,N为定理1所定义。则

i+(N)=2r(A1)+2r(B1)+m2。

i-(N)=2r(A1)+2r(B1)+m2。

特别的，在定理2中，当A2=Im2时有如下推论。

推论1矩阵A1,B1,C1如定理1所述。M3,N3如下定义：

则: a)A1XB1=C1存在满足X

b)A1XB1=C1存在满足X>C2的Hermitian最小二乘解当且仅当i-(N3)=2r(A1)+2r(B1)+m2；

c)A1XB1=C1存在满足X≤C2的Hermitian最小二乘解当且仅当r(M3)=i+(N3)-m2；

d)A1XB1=C1存在满足X≥C2的Hermitian最小二乘解当且仅当r(M3)=i-(N3)-m2。

如果在定理1中有A2=Im2,C2=0，则可得到矩阵方程A1XB1=C1存在(半)正(负)定Hermitian最小二乘解的等价条件如下。

推论2矩阵A1,B1,C1如定理1所述。M4,N4如下定义：

则: a)方程A1XB1=C1存在负定Hermitian最小二乘解当且仅当i+(N4)=2r(A1)+2r(B1)+m2；

b)方程A1XB1=C1存在正定Hermitian最小二乘解当且仅当i-(N4)=2r(A1)+2r(B1)+m2；

c)方程A1XB1=C1存在半负定Hermitian最小二乘解当且仅当r(M4)=i+(N4)-m2；

d)方程A1XB1=C1存在半正定Hermitian最小二乘解当且仅当r(M4)=i-(N4)-m2。

/

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WANG Fangyuan1, LI Ying1, XU Zhouhui2, CHA Xiuxiu1

(1.School of Mathematical Sciences,Liaocheng University,Liaocheng Shandong 252000,China;2.Department of Foundation Education,Technician College of Liaocheng City,Liaocheng Shandong 252000,China)

matrix function; matrix equation; rank; inertia; least-squares solution

2014-03-28；

2014-04-24；责任编辑：张 军

国家自然科学基金(11301247)

王方圆(1987-),女,河南许昌人,硕士研究生,主要从事线性系统理论方面的研究。

李 莹副教授。 E-mail:liyingld@163.com

1008-1542(2014)06-0529-09

10.7535/hbkd.2014yx06007

O151.21MSC(2010)主题分类15A60

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