AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 423605 10.1155/2013/423605 423605 Research Article The Hermitian R-Conjugate Generalized Procrustes Problem Chang Hai-Xia 1 Duan Xue-Feng 2 http://orcid.org/0000-0002-0494-4928 Wang Qing-Wen 3 Hajarian Masoud 1 Department of Applied Mathematics Shanghai Finance University Shanghai 201209 China shfc.edu.cn 2 School of Mathematics and Computational Science Guilin University of Electronic Technology Guilin 541004 China gliet.edu.cn 3 Department of Mathematics Shanghai University Shanghai 200444 China shu.edu.cn 2013 28 9 2013 2013 10 05 2013 14 08 2013 2013 Copyright © 2013 Hai-Xia Chang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We consider the Hermitian R-conjugate generalized Procrustes problem to find Hermitian R-conjugate matrix X such that k=1pAkX-Ck2 + l=1qXBl-Dl2 is minimum, where Ak, Ck, Bl, and Dl (k=1,2,,p, l=1,,q) are given complex matrices, and p and q are positive integers. The expression of the solution to Hermitian R-conjugate generalized Procrustes problem is derived. And the optimal approximation solution in the solution set for Hermitian R-conjugate generalized Procrustes problem to a given matrix is also obtained. Furthermore, we establish necessary and sufficient conditions for the existence and the formula for Hermitian R-conjugate solution to the linear system of complex matrix equations A1X=C1, A2X=C2,,ApX=Cp, XB1=D1,,XBq=Dq (p and q are positive integers). The representation of the corresponding optimal approximation problem is presented. Finally, an algorithm for solving two problems above is proposed, and the numerical examples show its feasibility.

1. Introduction

Throughout, let 𝒞m×n denote the set of all complex m×n matrices, m×n the set of all real m×n matrices, and rm×n the set of all matrices in m×n with rank r. The symbols I, A¯, AT, A*, A, and A, respectively, stand for the identity matrix with the appropriate size, the conjugate, the transpose, the conjugate transpose, the Moore-Penrose inverse, and the Frobenius norm of A𝒞m×n. For A=(aij), B=(bij)𝒞m×n; A*B=(aijbij)𝒞m×n represents the Hadamard product of A and B.

A linear model of image restoration is a matrix-vector equation is (1)y=Kx+n, where y represents the observed image, x the original true image, n additive noise, and K a blurring matrix. Image restoration is to minimize blur in an observed image, namely, recover a optimal approximation of x by given y and K, and get some statistical information of n. The process of the image restoration for the model (1) can be described as (2)minxx2subject  to  Kx-y2=ε, where ε is a small positive parameter. It is known that x^=Ky is the least squares solution of (1) with minimal norm. However, for ε=0, the solution x^ε to (2), that is, n=0 in (1) is not feasible. We know if ε0, then the solution x^ε converges to x^. In order to obtain a solution x^ε sufficiently near to x^, we usually take small ε such that 0<εδ, where the error norm δ:=n is given.

Now we consider the generalized problem of the process of the image restoration.

Problem 1.

Given 𝒮𝒞n×n, mk, hl, p, q being positive integers, Ak,Ck𝒞mk×n, Bl,Dl𝒞n×hl, E𝒞n×n, and k=1,,p, l=1,,q. Let (3)C={X𝒮k=1pAkX-Ck2+l=1qXBl-Dl2=min}. Find X^C such that (4)X^-E=minXCX-E.

The constraint Procrustes problem associated with several kinds of sets 𝒮, that is, p=1 and q=0 in (3) has been extensively studied, such as the orthogonal Procrustes problem with 𝒮 being the set of orthogonal matrices , the symmetric Procrustes problem , (M,N)-symmetric Procrustes problem , Hermitian, Hermitian R-symmetric and Hermitian R-skew-symmetric Procrustes problems , the Procrustes problems with 𝒮 constrained to the cone of symmetric positive semidefinite and symmetric elementwise matrices , and the generalized Procrustes analysis . The optimal approximation problem (4) is initially proposed in the processes of testing or revising given data. A preliminary estimate E of the unknown matrix X in C can be obtained from experimental observation values and the information of statistical distribution.

We characterize the case AkX=Ck, XBl=Dl, k=1,,p, l=1,,q in Problem 1 and describe it as follows.

Problem 2.

Given 𝒮𝒞n×n, m1,,mp, h1,,hq, p and q being positive integers, A1,C1𝒞m1×n, A2,C2𝒞m2×n,,Ap,Cp𝒞mp×n, B1,D1𝒞n×h1,,Bq,Dq𝒞n×hq, F𝒞n×n. Let (5)L={X𝒮A1X=C1,A2X=C2,,ApX=Cp,XB1=D1,,XBq=Dq}. When L is nonempty, find X~L such that (6)X~-F=minXLX-F.

For important results to solve Problem 2 with different sets 𝒮, we refer to .

Motivated by the work mentioned above, in this paper we mainly discuss the above two problems associated with S being the set of Hermitian R-conjugate matrices.

Recall that an n×n complex matrix K is R-conjugate if K¯=RKR, where Rn×n is a nontrivial involution, that is, R2=I, R±I, which was defined in . A matrix K𝒞n×n is Hermitian R-conjugate if K*=K and K¯=RKR, where RT=R-1=R±I. The Hermitian R-conjugate matrix is very useful in scientific computation and digital signal and image processing, its special case, for example, Hermitian Toeplitz matrix, have been studied by several authors, see [14, 1621]. We denote the set of all n×n Hermitian R-conjugate matrices by HRc𝒞n×n. Let Rc𝒞n×n, Sn×n, and ASn×n denote the set of all n×n complex R-conjugate matrices, real symmetric matrices, real skew-symmetric matrices, respectively.

This paper is organized as follows. In Section 2, we give some preliminary lemmas. In Section 3, we derive the expression of the unique solution to the Problem 1 with 𝒮=HRc𝒞n×n. In Section 4, we establish the solvability conditions for existence and an expression of the solution for Problem 2 with 𝒮=HRc𝒞n×n. In Section 5, we give examples to illustrate the results obtained in this paper.

2. Preliminaries

In order to study Problems 1 and 2 with 𝒮=HRc𝒞n×n, we first give some preliminary lemmas in this section.

For a nontrivial symmetric involutory matrix Rn×n, there exist positive integers r and s such that r+s=n and an n×n orthogonal matrix [P,Q] such that (7)R=[P,Q][Ir00-Is][PTQT], where Pn×r and Qn×s. The columns of P(Q) form an orthogonal basis for the eigenspace of R associated with the eigenvalue 1(-1).

Throughout this paper, we always suppose the nontrivial symmetric involutory matrix R is fixed which is given by (7).

Lemma 3 (see Theorem 2.1 in [<xref ref-type="bibr" rid="B21">14</xref>]).

A matrix KRc𝒞n×n if and only if there exists H1n×n such that K=UH1U*, and KHRc𝒞n×n if and only if there exists H2Sn×n such that K=UH2U*, where (8)U=[P,-iQ], with P and Q being the same as (7).

Lemma 4.

For any matrix A𝒞n×n, A=A1A2iA3, where (9)B=A+RA¯R2,A1=B+B*2,A2=B-B*2,A3=A-RA¯R2i, and denotes the direct sum.

Proof.

For B𝒞n×n, it is obvious that B=A1A2, where A1 are A2 are defined in (9). Hence, we just need to prove A=BiA3.

For any matrix A𝒞n×n, it is obvious that A=B+iA3, where B and A3 are defined in (9).

We prove the uniqueness of A=B+iA3. Note that (10)RBR=B¯,RA3R=A3¯, that is, B,A3Rc𝒞n×n. If there exist D and C3 satisfying (11)RDR=D¯,RC3R=C3¯ such that A=D+iC3, then (12)B-D=i(C3-A3). Multiplying R on the left and right side and then taking the conjugate for (12), it yields (13)B-D=i(A3-C3), implying B=D and A3=C3.

So A=A1A2iA3 holds, where A1, A2, and A3 are defined in (9).

Lemma 5.

Let the symmetric involution Rn×n be given in (7) and let U be defined as (8). Then

a matrix A𝒞m×n satisfies AR=A¯ if and only if AURm×n,

a matrix B𝒞n×l satisfies RB=B¯ if and only if  U*BRn×l.

Proof.

(i) It yields from (7) and (8) that (14)R=[P,-iQ][Ir00-Is][PTiQT]=U[Ir00-Is]U*. By (14) (15)A¯=AR=AU[Ir00-Is]U*, that is, (16)A=AU¯[Ir00-Is]U¯*=AU¯U*, implying AU=AU¯, that is, AURm×n.

Conversely, if AURm×n, according to the proof of the necessity, we can get A¯=AR.

The proof of (ii) can be analogously completed according to the proof of (i).

3. The Solution to Problem <xref ref-type="statement" rid="problem1">1</xref> with <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M198"><mml:mi>𝒮</mml:mi><mml:mo mathvariant="bold">=</mml:mo><mml:mi>H</mml:mi><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>𝒞</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo mathvariant="bold">×</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula>

We, in this section, give the explicit expression of the solution to Problem 1 with 𝒮=HRc𝒞n×n. In the following, we refer to the 𝒮=HRc𝒞n×n in C.

According to Lemma 3, if XHRc𝒞n×n, then (17)X=UYU*, where U is defined as (8) and YSn×n. Let (18)Ak=Ak1+iAk2𝒞mk×n,Ak1=Ak+Ak¯R2,Ak2=Ak-Ak¯R2i,(19)Bl=Bl1+iBl2𝒞n×hl,Bl1=Bl+RBl¯2,Bl2=Bl-RBl¯2i,(20)CkU=Ck1+iCk2,Ck1,Ck2mk×n,U*Dl=Dl1+iDl2,Dl1,Dl2n×hl. By Lemma 5, it is easy to verify Ak1U,Ak2Umk×n, U*Bl1,U*Bl2n×hl. We obtain (21)AkX-Ck=(Ak1+iAk2)UYU*-Ck=[(Ak1UY-Ck1)+i(Ak2UY-Ck2)]U*,(22)XBl-Dl=UYU*(Bl1+iBl2)-Dl=U[YU*(Bl1-Dl1)+iYU*(Bl2-Dl2)]. It follows from the unitary invariance of Frobenius norm, (21), (22), and YSn×n that (23)k=1pAkX-Ck2+l=1qXBl-Dl2=k=1p(Ak1UY-Ck12+Ak2UY-Ck22)+l=1q(YU*Bl1-Dl12+YU*Bl2-Dl22)=k=1p[Ak1UAk2U]Y-[Ck1Ck2]2+l=1qY[U*Bl1,U*Bl2]-[Dl1,Dl2]2=k=1p[Ak1UAk2U]Y-[Ck1Ck2]2+l=1q[(U*Bl1)T(U*Bl2)T]Y-[Dl1TDl2T]2. Suppose (24)S=[(A11U)T,(A12U)T,,(Ap1U)T,(Ap2U)T,U*B11,U*B12,,U*Bq1,U*Bq2]T,(25)G=[C11T,C12T,,Cp1T,Cp2T,D11,D12,,Dq1,Dq2]T, then (26)k=1pAkX-Ck2+l=1qXBl-Dl2=SY-G2.

We first give the following lemma which can be obtained by contrast with Lemma 2.1 in .

Lemma 6.

Given S,G(m1++mp+h1++hq)×n; therefore, the singular value decomposition (SVD) of Sr1(m1++mp+h1++hq)×n can be described as (27)S=W[Σr1000]VT=W1Dr1V1T, where (28)W=[W1,W2],V=[V1,V2], Let W1(m1++mp+h1++hq)×r1, W2(m1++mp+h1++hq)×(m1++mp+h1++hq-r1), V1n×r1, V2n×(n-r1), and Σr1=diag(d1,,dr1) with di>0, i=1,,r1; Φ=(1/(di2+dj2))r1×r1. Then minYSn×nSY-G is consistent if and only if (29)Y=V[Φ*(Σr1W1TGV1+V1TGTW1Σr1)Σr1-1W1TGV2V2TGTW1Σr1-1L]VT, where LS(n-r1)×(n-r1) is arbitrary.

Theorem 7.

Given Ak,Ck𝒞mk×n, Bl,Dl𝒞n×hl, and positive integers mk, hl, p, and q, where k=1,,p, l=1,,q, the notations Ak1, Ak2, Bl1, Bl2, Ck1, Ck2, Dl1, Dl2, S, G are defined as (18), (19), (20), (24), and (25), respectively. Let the SVD of Sr1(m1++mp+h1++hq)×n be of the form (27) with (28). Then (30)C={[Φ*(Σr1W1TGV1+V1TGTW1Σr1)Σr1-1W1TGV2V2TGTW1Σr1-1L]LS(n-r1)×(n-r1)UV×[Φ*(Σr1W1TGV1+V1TGTW1Σr1)Σr1-1W1TGV2V2TGTW1Σr1-1L]×VTU*[Φ*(Σr1W1TGV1+V1TGTW1Σr1)Σr1-1W1TGV2V2TGTW1Σr1-1L]}.

Proof.

It yields from (26) that (31)minXHRc𝒞n×nk=1pAkX-Ck2+l=1qXBl-Dl2=minYSn×nSY-G2. By Lemma 6, minYSn×nSY-G is consistent if and only if Y has the expression of (29). Taking (29) into (17), we obtain the solution set C is (30).

Theorem 8.

Given E𝒞n×n, the equation (4) is consistent if and only if (32)X^=UV×[Φ*(Σr1W1TGV1+V1TGTW1Σr1)Σr1-1W1TGV2V2TGTW1Σr1-1V2TU*(E+E*)UV22]×VTU*.

Proof.

Obviously, C is a closed convex set. Hence, there exists the unique element X^L such that (4) holds. By applying Theorem 7 and the unitary invariance of Frobenius norm, for XC, we get (33)X-E2=UV[Φ*(Σr1W1TGV1+V1TGTW1Σr1)Σr1-1W1TGV2V2TGTW1Σr1-1L]×VTU*-E[Φ*(Σr1W1TGV1+V1TGTW1Σr1)Σr1-1W1TGV2V2TGTW1Σr1-1L]2=[Φ*(Σr1W1TGV1+V1TGTW1Σr1)Σr1-1W1TGV2V2TGTW1Σr1-1L]-VTU*EUV[Φ*(Σr1W1TGV1+V1TGTW1Σr1)Σr1-1W1TGV2V2TGTW1Σr1-1L]2=Φ*(Σr1W1TGV1+V1TGTW1Σr1)-V1TU*EUV12+Σr1-1W1TGV2-V1TU*EUV22+V2TGTW1Σr1-1-V2TU*EUV12+L-V2TU*EUV22. Then minXCX-E is equivalent to (34)minLS(n-r1)×(n-r1)L-V2TU*EUV2. (34) holds if and only if (35)L=V2TU*(E+E*)UV22. Substituting (35) into XC, we get X^ is (32).

Corollary 9.

X min = min X C X if and only if (36)Xmin=UV×[Φ*(Σr1W1TGV1+V1TGTW1Σr1)Σr1-1W1TGV2V2TGTW1Σr1-10]×VTU*.

Remark 10.

when p=1 and q=1 in Theorem 8, we can derive a result of Theorem 4.1 in .

4. The Solution to Problem <xref ref-type="statement" rid="problem2">2</xref> with <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M269"><mml:mi>𝒮</mml:mi><mml:mo mathvariant="bold">=</mml:mo><mml:mi>H</mml:mi><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>𝒞</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo mathvariant="bold">×</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula>

We refer to 𝒮=HRc𝒞n×n in L in the following text. In this section, we establish necessary and sufficient conditions for the existence and the expression of L. When L is nonempty, we present the expression of the unique solution to (6).

It follows from (21) that the system A1X=C1, A2X=C2,,ApX=Cp, XB1=D1,,XBq=Dq with unknown XHRc𝒞n×n is consistent if and only if there exists YSn×n such that SY=G. We first give the following lemma.

Lemma 11 (see Theorem 1 in [<xref ref-type="bibr" rid="B8">7</xref>]).

Given S,GR(m1++mp+h1++hq)×n. Let the SVD of SRr1(m1++mp+h1++hq)×n be (27) with (28). Then the matrix equation SY=G has a symmetric solution if and only if (37)SSG=G,GST=SGT, and the symmetric solution can be expressed as (38)Y=SG+(I-SS)(SG)T+V2ZV2T, where ZS(n-r1)×(n-r1) is arbitrary.

Theorem 12.

Given A1,C1𝒞m1×n, A2,C2𝒞m2×n,,Ap,Cp𝒞mp×n, B1,D1𝒞n×h1,,Bq,Dq𝒞n×hq, and positive integers p and q. The notations Ak1, Ak2, Bl1, Bl2, Ck1, Ck2, Dl1, Dl2, S, G are defined as (18), (19), (20), (24), and (25), respectively. Let the SVD of Sr1(m1++mp+h1++hq)×n be of the form (27) with (28). Then the solution set L is nonempty if and only if (37) holds, in which case, (39)L={[SG+(I-SS)(SG)T+V2ZV2T]ZS(n-r1)×(n-r1)U[SG+(I-SS)(SG)T+V2ZV2T]U*}.

Proof.

If the solution set L is nonempty, then there exists a matrix XHRc𝒞n×n such that A1X=C1, A2X=C2,,ApX=Cp, XB1=D1,,XBq=Dq. We know that A1X=C1, A2X=C2,,ApX=Cp, XB1=D1,,XBq=Dq with XHRc𝒞n×n is consistent if and only if there exists YSn×n such that SY=G. By Lemma 11, SY=G is solvable for YSn×n if and only if (37) holds, and the expression of the solution is (38). Insert (38) into (17), then we obtain L is of the form (39).

Conversely, assume (37) holds, according to the proof of the necessity, A1X=C1, A2X=C2,,ApX=Cp, XB1=D1,,XBq=Dq is solvable for XHRc𝒞n×n. For ZS(n-r1)×(n-r1), by Lemma 3, U[SG+(I-SS)(SG)T+V2ZV2T]U*HRc𝒞n×n.

Theorem 13.

Given F𝒞n×n and L is nonempty. Let F1=(1/4)[F+RF¯R+(F+RF¯R)*]. Then (6) is consistent for X~L if and only if (40)X~=U[SG+(I-SS)(SG)T+V2V2TUF1U*V2V2T]U*. In particular, if Xinf=minXLX, then (41)Xinf=U[SG+(I-SS)(SG)T]U*.

Proof.

By Lemma 4, F=F1F2iF3, where (42)F1=14[F+RF¯R+(F+RF¯R)*],F2=14[F+RF¯R-(F+RF¯R)*],F3=F-RF¯R2i. When L is nonempty, for XL, we get (43)X-F2=U[SG+(I-SS)(SG)T+V2ZV2T]U*-(F1+F2+iF3)[SG+(I-SS)(SG)T+V2ZV2T]2=SG+(I-SS)(SG)T+V2ZV2T-U*F1U-U*F2U-iU*F3U(I-SS)(SG)T2. Since F1HRc𝒞n×n, F2,F3Rc𝒞n×n, by Lemma 3, we obtain U*F1USn×n, U*F2U,U*F3Un×n. Note that U*F2UASn×n, then (44)X-F2=UF1U*-SG-(I-SS)(SG)T-V2ZV2T2+UF2U*2+UF3U*2. Hence, minXLX-F is equivalent to (45)minZS(n-r1)×(n-r1)UF1U*-SG-(I-SS)(SG)T-V2ZV2T. For the orthogonal matrix V, we get (46)UF1U*-SG-(I-SS)(SG)T-V2ZV2T=VT[UF1U*-SG-(I-SS)(SG)T-V2ZV2T]V=V1T[UF1U*-SG-(I-SS)(SG)T]V1+V1T[UF1U*-SG-(I-SS)(SG)T]V2+V2T[UF1U*-SG-(I-SS)(SG)T]V1+V2T[UF1U*-SG-(I-SS)(SG)T]V2-Z. Then minZS(n-r1)×(n-r1)UF1U*-SG-(I-SS)(SG)T-V2ZV2T is solvable if and only if (47)Z=V2T[UF1U*-SG-(I-SS)(SG)T]V2. It follows from (27) that (48)S=V1Dr1-1W1T. By (48) and (47) we get (49)Z=V2TUF1U*V2. Hence, from (39) and (49), we obtain X~ which can be expressed as (40).

Remark 14.

When p=1 and q=1 in Theorem 13, we can get a conclusion of Theorem 3.1 in .

5. Numerical Examples

We, in this section, propose an algorithm for finding the solution of Problems 1 and 2 with 𝒮=HRc𝒞n×n and give illustrative numerical examples.

Let the symmetric involutory matrix R=[00V20I20V200]6×6, where V2=. Applying the spectral decomposition of R, we obtain the orthogonal matrix [P,Q] satisfying (7) and then by (8), we get (50)U=12[00100-i0001i02000000200000001-i000100i].

Algorithm 15.

( 1 ) Input A1,A2,,Ap, C1,C2,,Cp, B1,,Bq, D1,,Dq;

( 2 ) compute Ak1, Ak2, Bl1, Bl2, Ck1, Ck2, Dl1, Dl2, S, G by (18), (19), (20), (24), and (25);

( 3 ) make the SVD of S with the form of (27) and compute W1, W2, V1, V2 by (28);

( 4 ) if (37) holds, continue, or go to step 6;

( 5 ) input E𝒞n×n, compute the solution to Problem 2 with 𝒮=HRc𝒞n×n by (40);

( 6 ) input F𝒞n×n, compute the solution of Problem 1 with 𝒮=HRc𝒞n×n by (32).

Example 1.

We consider the case of p=1 and q=1. Suppose A,C𝒞5×6, B,D𝒞6×4, and(51)A=[-2.2163+1.2010i2.2524-1.1328i0-0.0955+1.3420i4.3215-2.3145i1.2120+1.9804i-1.16253.2510i02.1800-0.3125i-0.2235+1.4140i5.70711.0500-2.0120i2.4020+1.2500i03.2106i-1.60802.25600000003.2000-4.1955+1.2000i01.1025-0.2400i-0.8902-0.2408i1.2010+0.7071i],C=[4.1365+2.6614i1.5828-0.2259i2.2091+1.5678i2.4154i-1.8939+0.4692i2.5155-0.7604i1.0118-0.2844i-3.4235+1.5798i1.3798-0.7942i1.3050+2.9900i-2.7833+0.0592i1.8744-0.7889i0000002.1991-1.0648i-1.5298i5.6808+1.6029i-4.0150+2.1200i2.4387-1.9600i3.4200-2.6288i2.98832.4329-0.6405i1.4611-1.0503i3.7680-1.4980i-2.0266-0.9334i1.5756i],B=[0.5200+0.7200i-0.20000.4500+0.3600i0-0.3394-0.5091i0.1414-0.3536i-0.3200-0.6500i00.5940+0.2970i0.0707+0.0354i0.070000.0200i0.25000.3500+0.8400i0-0.3960-0.5091i0.1414+0.3536i-0.3200+0.2400i00.5940+0.2970i0.2121+0.0354i0.07070],D=[-0.3140+0.1069i0.4954-0.0615i0.7469+0.4876i-0.0662+0.0817i0.4255-0.5754i0.0348+0.5182i-0.2522+0.5751i-0.0105-0.0928i0.0883+0.0649i0.2201+0.0182i0.3133+0.3271i0.0076-0.0138i0.8675+0.7972i-0.3105-0.0837i-0.1461+0.2541i0.2620+0.0983i-0.4591+0.6268i-0.0405-0.6447i0.1517-0.6198i0.0256+0.2057i0.1439-0.2759i0.4722+0.1091i0.4952+0.3680i-0.0414-0.1178i].We can easily verify the solvability condition (37) does not holds. For given E𝒞6×6,(52)E=[0.4424+0.6274i-0.1765-0.0108i0.4006+0.3308i0.3069-0.1505+0.1262i1.200+0.0045i0.6374-0.0108i0.2246-0.2738i0.1509-0.4327i-0.0434-0.0915i0.1385-0.1800i0-0.4950-0.4172i0.1414-0.0354i-0.3818-0.1750i-0.2475+0.0619i-0.1630-0.0540i0.2032i-0.5129-0.4025i0.1956-0.0157i0.0868+0.5565i-0.1227-0.2382-0.0911i00.1878+0.3187i0.1789+0.3334i0.0271+0.7668i-0.0434+0.0915i-0.1024-0.0002+0.0150i-0.2404-0.6010i0.1414+0.0354i-0.2546-0.3341i-0.2475-0.0619i-0.0002-0.0150i0.0256],applying Algorithm 15, we get the following results:(53)X^=[-0.1949+0.0000i-0.0395+0.3097i0.1067+0.0353i0.6067-0.2888i0.0380-0.6272i-0.1218+0.1242i-0.0395-0.3097i-0.7354+0.0000i0.0606-0.1460i-0.1095+0.4086i0.6954-0.0593i0.0380-0.6272i0.1067-0.0353i0.0606+0.1460i0.00090.39570.0606-0.1460i0.1067+0.0353i0.6067+0.2888i-0.1095-0.4086i0.3957-0.1562-0.1095+0.4086i0.6067-0.2888i0.0380+0.6272i0.6954+0.0593i0.0606+0.1460i-0.1095-0.4086i-0.7354-0.0000i-0.0395+0.3097i-0.1218-0.1242i0.0380+0.6272i0.1067-0.0353i0.6067+0.2888i-0.0395-0.3097i-0.1949-0.0000i],(54)AX^-C2+X^B-D2=minXHRc6×6AX-C2+XB-D2=160.0357,X^-X^*=4.1192×10-16,X¯-RXR=0,X^-E=3.7397.

The following example is about Problem 2 with 𝒮=HRc𝒞6×6, p=1 and q=0. We list results of comparison of the solutions computed by Algorithm 15 and MATLAB procedure X=AC.

Example 2.

Let Wa𝒞8×8, Va𝒞6×6 be unitary matrices,(55)Wa=[-0.1590+0.2812i-0.1803-0.1822i0.4873+0.0557i0.1272+0.2856i:0.1159-0.1077i0.4983-0.2783i0.4127-0.2359i0.3195+0.2342i:0.2197-0.1364i0.1146-0.1466i0.2900+0.0440i-0.5309-0.0423i:0.0016-0.0047i0.0108-0.0047i-0.0257+0.0018i-0.0181+0.0126i:0.2576-0.0271i0.2935+0.2509i-0.3014+0.0304i0.3562+0.0778i:0.4283+0.1012i0.0858-0.1579i-0.1758-0.3913i-0.5198+0.0316i:0.3884-0.2103i0.2167+0.2285i0.1236-0.0447i0.1348-0.0089i:0.5625+0.1774i-0.2720-0.4682i-0.0047+0.3918i0.1951+0.0273i:-0.5270-0.1794i0.1734-0.2471i0.2466+0.1275i0.0437+0.1077i0.0562-0.0163i-0.2264+0.4255i-0.1461+0.0547i-0.0662-0.0426i-0.1748+0.4297i0.4788+0.1119i-0.1543-0.2114i0.0422-0.0388i-0.0011-0.0201i0.0249+0.0695i0.0506+0.2945i0.7638-0.5658i-0.3766-0.0582i0.3605-0.2131i-0.4447+0.1694i-0.0868-0.0677i-0.1860-0.3130i-0.2654-0.1257i0.0594+0.2924i-0.0976+0.0376i0.3112-0.2036i0.3628-0.1215i0.5860-0.0319i0.1016+0.1755i0.2087-0.1071i-0.0638-0.1313i-0.1736-0.2252i0.0555-0.0788i],Va=[0.46210.1052-0.6218-0.1836-0.58970.0852-0.2474-0.0088i-0.1896-0.5726i0.1951-0.1974i-0.5870-0.1096i-0.2811+0.1327i-0.2113-0.0036i0.4418-0.0309i-0.2001-0.2357i0.0004+0.2178i0.0696+0.1631i0.1926-0.3664i-0.6629-0.1362i-0.1299-0.4383i0.1856+0.1319i0.1214-0.4193i0.0858+0.5057i-0.2366-0.1016i-0.0917-0.4592i-0.2390+0.2258i-0.1938-0.3803i0.0533+0.3571i0.3007+0.3102i-0.3167-0.3755i0.3810-0.0794i0.4263+0.1739i0.3926-0.3760i0.3391-0.2301i0.2360+0.2699i-0.0153+0.2763i0.0807+0.3359i],and D(t)=[Dzeros(2,6)], where D=diag(3,2,1,t/1,t/2,t/3) and zeros(2,6) is a 2×6 zeros matrix. Let A(t)=WaD(t)Va* and XHRc𝒞6×6,(56)X=[0.2543-0.3625-1.1647i-0.1609-0.2793i0.8839-0.4914i0.5675-0.0043i-0.4143-0.4378i-0.3625+1.1647i-1.38860.0672-0.6470i-0.2316+0.2397i0.0524-0.4418i0.5675-0.0043i-0.1609+0.2793i0.0672+0.6470i-0.4312-0.25000.0672-0.6470i-0.1609-0.2793i0.8839+0.4914i-0.2316-0.2397i-0.2500-0.0380-0.2316+0.2397i0.8839-0.4914i0.5675+0.0043i0.0524+0.4418i0.0672+0.6470i-0.2316-0.2397i-1.3886-0.3625-1.1647i-0.4143+0.4378i0.5675+0.0043i-0.1609+0.2793i0.8839+0.4914i-0.3625+1.1647i0.2543].Then compute C(t)=A(t)X for t=10,1,10-1,10-2,,10-13. Obviously, Problem 2 with 𝒮=HRc𝒞n×n is consistent for each value of t. For matrices A, C obtained above, we first use Algorithm 15 to obtain the Hermitian R-conjugate solutions approximate to X, then compute the solutions of AX=C by MATLAB procedure X=AC. Let X~is denote the solutions computed by Algorithm 15 and Xis the solutions by MATLAB procedure X=AC. Let (57)e1=AX~i-C,e2=X~i-X,e3=X~i¯-RX~iR,e4=X~iT-X~i,te1=AXi-C,te2=Xi-X,te3=Xi¯-RXiR,te4=XiT-Xi.

Analysis of Results. As a general observation from Table 1, we find that the performance of Algorithm 15 to solve Problem 2 is very good and that of the MATLAB procedure X=AC is quite sensitive to the conditioning of matrix A. For t=10,1,,10-4, both methods behave well. In this case we should choose MATLAB procedure X=AC to solve Problem 2 with 𝒮=HRc𝒞n×n for it is simple. However, as some singular values of A are close to zero, the solutions Xi computed by MATLAB procedure X=AC do not satisfy AX=C and gradually lose the property of Hermitian R-conjugate matrix, while Algorithm 15 does it well. Hence, when A has small singular values close to zero, the Algorithm 15 predominates over MATLAB procedure X=AC.

Comparison of results by Algorithm 15 and the MATLAB procedure AC.

t e 1 e 2 e 3 e 4 t e 1 t e 2 t e 3 t e 4
- 13 2 · 1 0 - 15 1 · 1 0 - 15 0 2 · 1 0 - 15 3 · 1 0 - 15 3 · 1 0 - 2 5 · 1 0 - 2 4 · 1 0 - 2
- 12 2 · 1 0 - 15 1 · 1 0 - 15 0 1 · 1 0 - 15 3 · 1 0 - 15 3 · 1 0 - 3 5 · 1 0 - 3 4 · 1 0 - 3
- 11 2 · 1 0 - 15 2 · 1 0 - 15 0 1 · 1 0 - 15 3 · 1 0 - 15 3 · 1 0 - 4 4 · 1 0 - 4 5 · 1 0 - 4
- 10 2 · 1 0 - 15 1 · 1 0 - 15 0 2 · 1 0 - 15 3 · 1 0 - 15 3 · 1 0 - 5 3 · 1 0 - 5 4 · 1 0 - 5
- 9 3 · 1 0 - 15 2 · 1 0 - 15 0 2 · 1 0 - 15 2 · 1 0 - 15 3 · 1 0 - 6 5 · 1 0 - 6 4 · 1 0 - 6
- 8 2 · 1 0 - 15 1 · 1 0 - 15 0 2 · 1 0 - 15 2 · 1 0 - 15 3 · 1 0 - 7 4 · 1 0 - 7 3 · 1 0 - 7
- 7 3 · 1 0 - 15 2 · 1 0 - 15 0 1 · 1 0 - 15 2 · 1 0 - 15 4 · 1 0 - 8 5 · 1 0 - 8 5 · 1 0 - 8
- 6 2 · 1 0 - 15 2 · 1 0 - 15 0 2 · 1 0 - 15 2 · 1 0 - 15 2 · 1 0 - 9 4 · 1 0 - 9 4 · 1 0 - 9
- 5 2 · 1 0 - 15 1 · 1 0 - 15 0 2 · 1 0 - 15 2 · 1 0 - 15 2 · 1 0 - 10 3 · 1 0 - 10 3 · 1 0 - 10
- 4 2 · 1 0 - 15 1 · 1 0 - 15 0 1 · 1 0 - 15 2 · 1 0 - 15 3 · 1 0 - 11 3 · 1 0 - 11 4 · 1 0 - 11
- 3 2 · 1 0 - 15 1 · 1 0 - 15 0 2 · 1 0 - 15 3 · 1 0 - 15 3 · 1 0 - 12 5 · 1 0 - 12 5 · 1 0 - 12
- 2 2 · 1 0 - 15 1 · 1 0 - 15 0 2 · 1 0 - 15 2 · 1 0 - 15 2 · 1 0 - 13 3 · 1 0 - 13 3 · 1 0 - 13
- 1 2 · 1 0 - 15 1 · 1 0 - 15 0 2 · 1 0 - 15 3 · 1 0 - 15 3 · 1 0 - 14 5 · 1 0 - 14 5 · 1 0 - 14
0 2 · 1 0 - 15 2 · 1 0 - 15 0 2 · 1 0 - 15 4 · 1 0 - 15 4 · 1 0 - 15 6 · 1 0 - 15 6 · 1 0 - 15
1 9 · 1 0 - 15 2 · 1 0 - 15 0 2 · 1 0 - 15 1 · 1 0 - 14 5 · 1 0 - 15 6 · 1 0 - 15 8 · 1 0 - 15
6. Conclusion

In this paper, we converted the Hermitian R-conjugate generalized Procrustes problem to real symmetric Procrustes problem trickly and obtained its solution set. We also investigated the Hermitian R-conjugate solution to the linear system of complex matrix equations A1X=C1, A2X=C2,,ApX=Cp, XB1=D1,,XBq=Dq and established solvable conditions and the formula for the Hermitian R-conjugate solution. Moreover, we showed the optimal approximation solution to a given matrix in the above two corresponding solution set is unique, respectively. As applications, a numerical algorithm has been given and the examples have illustrated the feasibility of the algorithm. Additionally, we can further consider the least square Hermitian R-conjugate solution of the system AkXAk*=Ck, k=1,2,,n (n is positive integer) and the corresponding optimal approximation problem.

Acknowledgments

The authors are very much indebted to the anonymous referees for their constructive and valuable comments and suggestions which greatly improved the original paper. This research was supported by the Innovation Foundation of Shanghai Municipal Education Commission (13ZZ080), the National Natural Science Foundation of China (11391240185), the Shanghai University Scientific Selection and Cultivation for Outstanding Young Teachers in special fund (sjr10009), and the Natural Science Foundation of Guangxi Province (no. 2012GXNSFBA053006).

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