We present the *congruence class of the least-square and the minimum norm least-square solutions to the system of complex matrix equation AX=C,XB=D by generalized singular value decomposition and canonical correlation decomposition.

1. Introduction

Throughout we denote the complex m×n matrix space by Cm×n. The symbols I, A*, and ∥A∥ stand for the identity matrix with the appropriate size, the conjugate transpose, and the Frobenius norm of A∈Cm×n, respectively. Recall that matrices X, Y∈Cn×n are in the same *congruence class if there is a nonsingular P∈Cn×n such that X=P*YP [1].

Investigating the classical system of matrix equations
(1)AX=C,XB=D
has attracted many people’s attention and many results have been obtained about system (1) with various constraints, such as Hermitian, positive definite, positive semidefinite, reflexive, and generalized reflexive solutions (see [2–10]). Studying the least-square solutions of the system of matrix equations (1) is also a very active research topic (see [11–16]). It is well known that Hermitian, positive definite and positive semidefinite matrices are the special case of *congruence. Therefore investigating the *congruence class of a solution of the matrix equation (1) is very meaningful.

In 2005, Horn et al. [1] studied the possible *congruence class of a square solution when linear matrix equation AX=B is solvable. In 2009, Zheng et al. [17] describe *congruence class of least-square and minimum norm least-square solutions of the equation AX=B when it is not solvable and discuss a *congruence class of the solutions of the system (1) when it is solvable. To our knowledge, so far there has been little investigation of *congruence class of the least-square and minimum norm least-square solutions to (1) when it is not solvable.

Motivated by the work mentioned above, we investigate the *congruence class of the least-square and the minimum norm least-square solutions to the system of complex matrix equation (1) by generalized singular value decomposition (GSVD) and canonical correlation decomposition (CCD).

2. The <inline-formula>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M25">
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</mml:math></inline-formula>Congruence Class of the Solutions to (<xref ref-type="disp-formula" rid="EEq1.1">1</xref>)Lemma 1 (see [<xref ref-type="bibr" rid="B4">4</xref>]).

Let A∈Cm×n and B∈Cn×p. Then the GSVD of A and B* can be expressed as
(2)A=UΣAP,B*=VΣBP,
where U∈Cm×m and V∈Cp×p are unitary matrices, P∈Cn×n is nonsingular matrix,
(3)ΣA∈Cm×n,ΣB∈Cp×n,r=rank(AB*),ΣA=(IA·SA·0OA·)tsr-s-tn-r,ΣB=(OB·SB·0IB·)tsr-s-tn-r,IA and IB are identity matrices, OA and OB are zero matrices, and
(4)SA=diag(α1,…,αs),SB=diag(β1,…,βs)
with 1>α1≥⋯≥αs>0, 0<β1≤⋯≤βs<1, and αi2+βi2=1(i=1,…,s).

For convenience, in the following theorem we denote(5)PXP*=(X11X12X13X14X21X22X23X24X31X32X33X34X41X42X43X44),tsr-s-tn-r(6)U*CP*=(C11C12C13C14C21C22C23C24C31C32C33C34),tsr-s-tn-rPDV=(D11D12D13D21D22D23D31D32D33D41D42D43)p-r+tsr-s-t.

Theorem 2.

Let A, C∈Cm×n, B, D∈Cn×p, and the GSVD of A and B* be expressed as (2), and then one has the following.

The system of matrix equation (1) has a solution in Cn×n if and only if
(7)C3i=0,Di1=0,(i=1,2,3,4),C12=D12SB-1,C13=D13,SA-1C22=D22SB-1,SA-1C23=D23.

In that case, the general solutions of (1) are
(8)X=P-1(C11C12C13C14SA-1C21SA-1C22D23SA-1C24X31D32SB-1D33X34X41D42SB-1D43X44)(P-1)*,

where X31, X41, X34, and X44 are arbitrary.

For arbitrary X31, X41, X34, and X44, there exists a solution in Cn×n of (1) which is *congruent to
(9)Y=(C11C12C13C14SA-1C21SA-1C22D23SA-1C24X31D32SB-1D33X34X41D42SB-1D43X44).

There exists a minimum norm solution in Cn×n of (1) which is *congruent to
(10)Y=(C11C12C13C14SA-1C21SA-1C22D23SA-1C240D32SB-1D3300D42SB-1D430).

Proof.

Using the GSVD of A and B* given by (2), we get
(11)AX=C⟺UΣAPX=C⟺ΣAPXP*=U*CP*,XB=D⟺XP*ΣB*V*=D⟺PXP*ΣB*=PDV.
By (2) and (5), ΣAPXP* and PXP*ΣB* have the following matrix decomposition:
(12)ΣAPXP*=(X11X12X13X14SAX21SAX22SAX23SAX240000),PXP*ΣB*=(0X12SBX130X22SBX230X32SBX330X42SBX43),
and we have that system (1) is equivalent to
(13)(X11X12X13X14SAX21SAX22SAX23SAX240000)=(C11C12C13C14C21C22C23C24C31C32C33C34),(0X12SBX130X22SBX230X32SBX330X42SBX43)=(D11D12D13D21D22D23D31D32D33D41D42D43);
obviously, the system of matrix equation (1) has a solution in Cn×n if and only if
(14)C3i=0,Di1=0,SAX2i=C2i,Xi2SB=Di2,X1i=C1i,Xi3=Di3,vvvvvvvvvvvvvvvvvvvv(i=1,2,3,4).
Therefore, (1) has a solution in Cn×n if and only if (7) holds, and a general form of the solutions can be expressed as (8); for arbitrary X31, X41, X34, and X44, there exists a solution in Cn×n of (1) which is *congruent to (9), and the part (d) follows from the definition of Frobenius norm.

Remark 3.

In 2009, Zheng et al. [17] discuss a *congruence class of the solutions of the system (1) when it is solvable. Our result in Theorem 2 is different with the result mentioned above.

3. The <inline-formula>
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</mml:math></inline-formula>Congruence Class of the Least-Square Solutions to (<xref ref-type="disp-formula" rid="EEq1.1">1</xref>)Lemma 4 (see [<xref ref-type="bibr" rid="B18">18</xref>]).

Let the CCD of matrix pair [A,C] with A∈Cm×n, C∈Cm×k, rank A=g, and rank C=h be given as
(15)A=U(ΣA,0)EA-1,C=U(ΣC,0)EC-1,
where U is a unitary matrix and
(16)ΣA=(IiΛj0¯-0_ΔjIt),ΣC=(Ih0¯)
are nonsingular matrices with the same row partitioning, and g=i+j+t,
(17)Λj=diag(λi+1,…,λi+j),1>λi+1≥⋯≥λi+j>0,Δj=diag(Δi+1,…,Δi+j),0>Δi+1≥⋯≥Δi+j>1,Λj2+Δj2=Ij,U=(u1u2u3u4u5u6)ijh-i-jm-h-j-tjt.

Lemma 5 (see [<xref ref-type="bibr" rid="B18">18</xref>]).

Given E, F∈Cm×n, then there exists a unique matrix S¯∈Cm×n such that
(18)∥S-E∥2+∥S-F∥2=min,
and S¯ can be expressed as
(19)S¯=E+F.

Lemma 6 (see [<xref ref-type="bibr" rid="B10">10</xref>]).

Given E, F∈Cm×n, Ω1=diag(a1,…,am), Ω2=diag(b1,…,bn), ai>0(i=1,…,m), and bj>0(j=1,…,n), then there exists a unique matrix S¯∈Cm×n such that
(20)∥Ω1S-E∥2+∥SΩ2-F∥2=min,
and S¯ can be expressed as
(21)S¯=Φ*(Ω1E+FΩ2),
where
(22)Φ=(1ai2+bj2)∈Cm×n.

Using Lemmas 5 and 6, we can easily obtain the following.

Lemma 7.

Given E, F, G∈Cm×n, Ω1=diag(a1,…,am), Ω2=diag(b1,…,bm), In=diag(i1,…,in), ai>0(i=1,…,m), bj>0(j=1,…,m), and ik=1(k=1,…,n), then there exist unique matrices S and W such that
(23)∥Ω1S+Ω2W-E∥2+∥S-F∥2+∥W-G∥2=min,
and S and W can be expressed as
(24)S=F,W=Φ*(Ω2(Ω1F-E)+G),
where
(25)Φ=(1bj2+ik2)∈Cm×n.

Lemma 8.

Given E, F∈Cm×n, Ω1=diag(a1,…,am), Ω2=diag(b1,…,bn), ai>0(i=1,…,m), and bj>0(j=1,…,n), then there exist unique matrices S and W such that
(26)∥Ω1S+Ω2W-E∥2=min,
and S and W can be expressed as
(27)S=0,W=Ω2-1E.

Let A, C∈Cm×n, B, D∈Cn×l, and rank A=p≥rank B=q. According to Lemma 4, there exist a unitary matrix U∈Cn×n and nonsingular matrices RA∈Cm×m and RB∈Cl×l, such that the CCD of matrix pair [A*,B] is given as
(28)A*=U(ΣA,0)RA-1,B=U(ΣB,0)RB-1,
where ΣA∈Cn×p, ΣB∈Cn×q,
(29)ΣA=(Ir000Gs00000000Ss000It),ΣB=(Ir000Is000Iq-r-s000000000),
where p=r+s+t,
(30)Gs=diag(gr+1,…,gr+s),1>gr+1≥⋯≥gr+s>0,Ss=diag(wr+1,…,wr+s),0>wr+1≥⋯≥wr+s>1,Gs2+Ss2=Is,U=(u1u2u3u4u5u6)rsq-r-sn-q-s-tst.
Without loss of generality, let p=q, and then we have the following results.

Theorem 9.

Let A, C∈Cm×n, B, D∈Cn×l, and the CCD of matrix pair [A*,B] be expressed as (28), and then one has the following.

The least-square solutions to the system (1) are(31)X=U(C11+D11C12+D12C13+D13C14C15C16D21D22D23000D31D32D33X34X35X36D41D42D43X44X45X46Y51Y52Y53S-1C24S-1C25S-1C26D31+D61D32+D62D33+D63C34C35C36)U*,

where X34, X35, X36, X44, X45, and X46 are arbitrary, Y5i=Φ*(S(GD2i-C2i)+D5i), i=1,2,3, Φ=(1/(wr+j2+ek2))∈Cs×s, and ek=1, j=1,…,s, k=1,…,s.

For arbitrary X34, X35, X36, X44, X45, and X46, there exists a least-square solution in Cn×n of (1) which is *congruent to(32)Y=(C11+D11C12+D12C13+D13C14C15C16D21D22D23000D31D32D33X34X35X36D41D42D43X44X45X46Y51Y52Y53S-1C24S-1C25S-1C26D31+D61D32+D62D33+D63C34C35C36),

where Y5i=Φ*(S(GD2i-C2i)+D5i), i=1,2,3, Φ=(1/(wr+j2+ek2))∈Cs×s, and ek=1, j=1,…,s, k=1,…,s.

There exists a minimum norm least-square solution in Cn×n of (1) which is *congruent to(33)Y=(C11+D11C12+D12C13+D13C14C15C16D21D22D23000D31D32D33000D41D42D43000Y51Y52Y53S-1C24S-1C25S-1C26D31+D61D32+D62D33+D63C34C35C36),

where Y5i=Φ*(S(GD2i-C2i)+D5i), i=1,2,3, Φ=(1/(wr+j2+ek2))∈Cs×s, and ek=1, j=1,…,s, k=1,…,s.Proof.

It follows from (28) that
(34)AX=C⟺(RA-1)*(ΣA*0)U*X=C⟺(ΣA*0)U*X=(RA)*C,XB=D⟺XU(ΣB,0)RB-1=D⟺XU(ΣB,0)=DRB.
Then,
(35)∥AX-C∥2+∥XB-D∥2=∥(ΣA*0)U*X-(RA)*C∥2+∥XU(ΣB,0)-DRB∥2=∥(ΣA*0)U*XU-(RA)*CU∥2+∥U*XU(ΣB,0)-U*DRB∥2.
Assume that
(36)U*XU=(X11X12X13X14X15X16X21X22X23X24X25X26X31X32X33X34X35X36X41X42X43X44X45X46X51X52X53X54X55X56X61X62X63X64X65X66),(37)(RA)*CU=(C11C12C13C14C15C16C21C22C23C24C25C26C31C32C33C34C35C36C41C42C43C44C45C46),U*DRB=(D11D12D13D14D21D22D23D24D31D32D33D34D41D42D43D44D51D52D53D54D61D62D63D64),
and then
(38)∥(ΣA*0)U*XU-(RA)*CU∥2+∥U*XU(ΣB,0)-U*DRB∥2=∥(X11X12⋯X16GX21+SX51GX22+SX52⋯GX26+SX56X61X62⋯X6600⋯0)-(C11C12⋯C16C21C22⋯C26C31C32⋯C36C41C42⋯C46)∥2+∥(X11X12X130X21X22X230⋮⋮⋮⋮X61X62X630)-(D11D12D13D14D21D22D23D24⋮⋮⋮⋮D61D62D63D64)∥2=∥X11-C11∥2+∥X11-D11∥2+∥X12-C12∥2+∥X12-D12∥2+∥X13-C13∥2+∥X13-D13∥2+∥X61-C31∥2+∥X61-D61∥2+∥X62-C32∥2+∥X62-D62∥2+∥X63-C33∥2+∥X63-D63∥2+∥GX21+SX51-C21∥2+∥X21-D21∥2+∥X51-D51∥2+∥GX22+SX52-C22∥2+∥X22-D22∥2+∥X52-D52∥2+∥GX23+SX53-C23∥2+∥X23-D23∥2+∥X53-D53∥2+∥GX24+SX54-C24∥2+∥GX25+SX55-C25∥2+∥GX26+SX56-C26∥2+∥X14-C14∥2+∥X15-C15∥2+∥X16-C16∥2+∥X64-C34∥2+∥X65-C35∥2+∥X66-C36∥2+∥X31-D31∥2+∥X32-D32∥2+∥X33-D33∥2+∥X41-D41∥2+∥X42-D42∥2+∥X43-D43∥2.
By Lemmas 5, 7, and 8, a general form of the least-square solutions can be expressed as (31); for arbitrary X34, X35, X36, X44, X45, and X46, there exists a least-square solution in Cn×n of (1) which is *congruent to (32), and the part (c) follows from the definition of Frobenius norm.

4. An Algorithm and Numerical Examples

Based on the main results of this paper, we in this section propose an algorithm for finding the least-square solutions to the system (1). All the tests are performed by MATLAB 6.5 which has a machine precision of around 10-16.

Algorithm 1.

(1) Input A∈Cm×nand B∈Cn×l, and compute U∈Cn×n, RA-1∈Cm×m, RB-1∈Cl×l, ΣA, ΣB∈Cn×p, and G,S∈Cs×s by the CCD of matrix pair [A*,B].

(2) Input C∈Cm×n, D∈Cn×l, and compute Cij(i=1,2,3,4;j=1,2,3,4,5,6) and Dlk(l=1,2,3,4,5,6;k=1,2,3,4) according to (37).

(3) Compute the least-square solutions of the system (1) by (31).

(4) Compute the *congruence class of the least-square and the minimum norm least-square solutions to the system (1) according to (32) and (33).

Applying Algorithm 1, we obtain the following:
(40)U=[10000000-i0000i000000000i000010000-i00],ΣA=[10000.5000000000.250001],ΣB=[100010001000000000],RA-1=[-1.625-2-0.752.6251.37510.25-1.375-0.8750-0.250.8750.37500.25-0.375],RB-1=[2-20-131-371-155-60-2-4452-121],G=[0.5],S=[0.25],(RA)*CU=(53i-3i-9i45i14-1+6i1-6i-19i111+10i32-2+26i2-23i-22i252+25i53-1+57i1-50i-34i381+46i),U*DRB=(0.03-0.05i0.02-0.05i0.04-0.02i0.02-0.04i-0.51i-0.27i-0.33i-0.25i0.43i0.21i0.29i0.2i0.41i0.26i0.25i0.23i1.350.670.780.62-1.27i-0.71i-0.82i-0.65i).

The least-square solutions to the system (1) are(41)X=U(5.03-0.05i0.02+2.95i0.04-3.02i-9i45i-0.51i-0.27i-0.33i0000.43i0.21i0.29iX34X35X360.41i0.26i0.25iX44X45X462.02-0.06i0.864-0.366i0.498+1.448i76i444+40i-0.84i-0.5i-0.53i-22i252+25i)U*,where X34, X35, X36, X44, X45, and X46 are arbitrary.

For arbitrary X34, X35, X36, X44, X45, and X46, there exists a least-square solution in C6×6 of (1) which is *congruent to(42)Y=(5.03-0.05i0.02+2.95i0.04-3.02i-9i45i-0.51i-0.27i-0.33i0000.43i0.21i0.29iX34X35X360.41i0.26i0.25iX44X45X462.02-0.06i0.864-0.366i0.498+1.448i76i444+40i-0.84i-0.5i-0.53i-22i252+25i).There exists a minimum norm least-square solution in C6×6 of (1) which is c*ongruent to(43)Y=(5.03-0.05i0.02+2.95i0.04-3.02i-9i45i-0.51i-0.27i-0.33i0000.43i0.21i0.29i0000.41i0.26i0.25i0002.02-0.06i0.864-0.366i0.498+1.448i76i444+40i-0.84i-0.5i-0.53i-22i252+25i).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research was supported by the Youth Funds of Natural Science Foundation of Hebei province (A2012403013), the Education Department Foundation of Hebei province (Z2013110), and the Natural Science Foundation of Hebei province (A2012205028).

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