矩陣方程的約束Hermitian最小二乘解

王方圓,李 瑩,許洲慧,查秀秀

(1.聊城大學(xué)數(shù)學(xué)科學(xué)學(xué)院,山東聊城 252000;2.聊城市技師學(xué)院基礎(chǔ)教學(xué)部，山東聊城 252000)

王方圓1,李 瑩1,許洲慧2,查秀秀1

(1.聊城大學(xué)數(shù)學(xué)科學(xué)學(xué)院,山東聊城 252000;2.聊城市技師學(xué)院基礎(chǔ)教學(xué)部，山東聊城 252000)

矩陣函數(shù)；矩陣方程；秩；慣性指數(shù)；最小二乘解

1 預(yù)備知識(shí)

引理2[10]設(shè)A∈Cm×n，B∈Cm×k,C∈Cl×n。則:

由引理2可得到以下秩公式：

引理3[11]設(shè)A∈Cm×n,B∈Cm×k,C∈Cl×n。則有：

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

b)[12]設(shè)A,B,C,D,P與Q使得矩陣表達(dá)式D-CP+AQ+B有意義。則有：

引理7[13]給定A∈Cm×n,B∈Cn×p與C∈Cm×p,設(shè)X∈Cn×n為變量矩陣,假設(shè)矩陣方程AXB=C相容，則下列各項(xiàng)等價(jià)。

a)矩陣方程AXB=C存在Hermitian解;

c)矩陣方程AYB=C與B*YA*=C*有公共解Y。

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

其中V∈Cn×n為任意矩陣。

記

則

2 約束Hermitian最小二乘解

由文獻(xiàn)[6]知A1XB1=C1的Hermitian最小二乘解的通解表達(dá)式為

(1)

(2)

式(2)是一個(gè)關(guān)于3個(gè)變量U1,U2與U3的線性Hermitian矩陣函數(shù)，將其表示為

(3)

(4)

對(duì)式(3)應(yīng)用引理9可得：

則

(5)

(6)

(7)

證明在式(4)的條件下,對(duì)式(3)應(yīng)用引理9得：

(8)

利用矩陣的初等變換可得：

(9)

(10)

利用引理2—引理6及引理9計(jì)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)

將式(9)—式(15)分別代入式(8)即得式(5)—式(7)。

利用以上秩與慣性指數(shù)極值與引理1即得如下結(jié)論。

定理2矩陣A1,B1,C1,A2,C2如定理1所述,記M1,M2,N1,N2,M,G,N為定理1所定義。則

i+(N)=2r(A1)+2r(B1)+m2。

i-(N)=2r(A1)+2r(B1)+m2。

特別的，在定理2中，當(dāng)A2=Im2時(shí)有如下推論。

推論1矩陣A1,B1,C1如定理1所述。M3,N3如下定義：

則: a)A1XB1=C1存在滿足X

b)A1XB1=C1存在滿足X>C2的Hermitian最小二乘解當(dāng)且僅當(dāng)i-(N3)=2r(A1)+2r(B1)+m2；

c)A1XB1=C1存在滿足X≤C2的Hermitian最小二乘解當(dāng)且僅當(dāng)r(M3)=i+(N3)-m2；

d)A1XB1=C1存在滿足X≥C2的Hermitian最小二乘解當(dāng)且僅當(dāng)r(M3)=i-(N3)-m2。

如果在定理1中有A2=Im2,C2=0，則可得到矩陣方程A1XB1=C1存在(半)正(負(fù))定Hermitian最小二乘解的等價(jià)條件如下。

推論2矩陣A1,B1,C1如定理1所述。M4,N4如下定義：

則: a)方程A1XB1=C1存在負(fù)定Hermitian最小二乘解當(dāng)且僅當(dāng)i+(N4)=2r(A1)+2r(B1)+m2；

b)方程A1XB1=C1存在正定Hermitian最小二乘解當(dāng)且僅當(dāng)i-(N4)=2r(A1)+2r(B1)+m2；

c)方程A1XB1=C1存在半負(fù)定Hermitian最小二乘解當(dāng)且僅當(dāng)r(M4)=i+(N4)-m2；

d)方程A1XB1=C1存在半正定Hermitian最小二乘解當(dāng)且僅當(dāng)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；責(zé)任編輯：張 軍

國(guó)家自然科學(xué)基金(11301247)

王方圓(1987-),女,河南許昌人,碩士研究生,主要從事線性系統(tǒng)理論方面的研究。

李 瑩副教授。 E-mail:liyingld@163.com

1008-1542(2014)06-0529-09

10.7535/hbkd.2014yx06007

O151.21MSC(2010)主題分類15A60

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