The-general-coupled-matrix-equations-over-generalized-bisymmetric-matrices_

Linear Algebra and its Applications432(2010)1531–1552Contents lists available at ScienceDirect Linear Algebra and its Applications j o u r n a l h o m e p a g e:w w w.e l s e v i e r.c o m/l o c a t e/l aaThe general coupled matrix equations over generalized bisymmetric matricesMehdi Dehghan∗,Masoud HajarianDepartment of Applied Mathematics,Faculty of Mathematics and Computer Science,Amirkabir University of Technology, No.424,Hafez Avenue,Tehran15914,IranA R T I C L E I N F O AB S T R AC TArticle history:Received20May2009Accepted12November2009 Available online16December2009 Submitted by S.M.RumpAMS classification:15A2465F1065F30Keywords:Iterative methodLeast Frobenius norm solution group Optimal approximation generalized bisymmetric solution group Generalized bisymmetric matrix General coupled matrix equations In the present paper,by extending the idea of conjugate gradient (CG)method,we construct an iterative method to solve the general coupled matrix equationspj=1A ij X jB ij=M i,i=1,2,...,p,(including the generalized(coupled)Lyapunov and Sylvester matrix equations as special cases)over generalized bisymmetric matrix group(X1,X2,...,X p).By using the iterative method,the solvability of the general coupled matrix equations over generalized bisym-metric matrix group can be determined in the absence of roundoff errors.When the general coupled matrix equations are consistent over generalized bisymmetric matrices,a generalized bisymmetric solution group can be obtained withinfinite iteration steps in the absence of roundoff errors.The least Frobenius norm generalized bisymmetric solution group of the general coupled matrix equa-tions can be derived when an appropriate initial iterative matrix group is chosen.In addition,the optimal approximation generalized bisymmetric solution group to a given matrix group( X1, X2,..., X p) in Frobenius norm can be obtained byfinding the least Frobe-nius norm generalized bisymmetric solution group of new general coupled matrix equations.The numerical results indicate that the iterative method works quite well in practice.©2009Elsevier Inc.All rights reserved.∗Corresponding author.E-mail addresses:mdehghan@aut.ac.ir,mdehghan.aut@(M.Dehghan),mhajarian@aut.ac.ir,masoudhajarian@ (M.Hajarian).0024-3795/$-see front matter©2009Elsevier Inc.All rights reserved.doi:10.1016/a.2009.11.0141532M.Dehghan,M.Hajarian /Linear Algebra and its Applications 432(2010)1531–15521.IntroductionFirst some symbols and notations are introduced.Let R m ×n be the set of all m ×n real matrices and SOR n ×n be the set of all symmetric orthogonal matrices in R n ×n .We denote by I n the n ×n identity matrix.We also write it as I ,when the dimension of this matrix is clear.The symbols A T ,tr (A )and R (A )stand the transpose,trace and column space of the matrix A ,respectively. A,B =tr (B T A )is defined as the inner product of the two matrices,which generates the Frobenius norm,i.e.||A ||2F =A,A =tr (A TA ).The Kronecker product of two matrices A andB is denoted by A ⊗B .The stretchingfunction vec (A )is defined as vec (A )= a T 1a T 2...a Tm T ,where a k is the k th column of A .The generalized bisymmetric matrices have wide applications in many fields which can be defined as follows:Definition 1.1.For arbitrary given matrix R ∈SOR n ×n ,i.e.,R =R T =R −1,we say that matrix A ∈R n ×n is generalized bisymmetric matrix with respect to R ,if RAR =A =A T .The set of order n gener-alized bisymmetric matrices with respect to R is denoted by BSR n ×nR .It is obvious that any symmetric matrix is also a generalized bisymmetric matrix with respect to I .In the literature,the problem for determining a solution to several linear matrix equations has been widely studied [10,14,22–24,27–30,33,35,43,44].In [4],the symmetric solution of the linear matrix equationAX=B,(1.1)have been considered using the singular-value,generalized singular-value,real Schur,and real gen-eralized Schur decompositions.By applying a formula for the partitioned minimum-norm reflexive generalized,Don [5]presented the general symmetric solution X to the matrix equation (1.1).By using the singular-value decomposition and the generalized singular-value decomposition,Dai [2]proposed the necessary and sufficient conditions for the consistency of two matrix equationsAX=C and AXB =C,(1.2)with a symmetric condition on solutions.In [1],the necessary and sufficient conditions for the existence of and the expressions for the symmetric solutions of the matrix equationsAX+YA =C,(1.3)AXA T +BYB T =C,(1.4)and(A T XA,B T XB )=(C,D ),(1.5)were derived.Many problems in systems and control theory require the solution of the matrix equation AX −YB =C .Baksalary and Kala [6]presented a condition for the existence of a solution and derived a formula for the general solution of the matrix equationAXB +CYD=E,(1.6)where A,B,C,D and E are given matrices of suitable sizes defined over real number field.Dehghan and Hajarian proposed an iterative method for solving a pair of matrix equations AYB =E and CYD =F over generalized centro-symmetric matrices [7]and some finite iterative algorithms for the reflexive and anti-reflexive solutions of (coupled)Sylvester matrix equations [8,9,13],and studied the lower bound for the product of eigenvalues of solutions to a class of linear matrix equations [12].In [26],the authors presented the necessary and sufficient conditions for the existence of constant solutions with bi(skew)symmetric constrains to the matrix equationsA i X−YB i =C i ,i =1,2,...,s,(1.7)M.Dehghan,M.Hajarian/Linear Algebra and its Applications432(2010)1531–15521533andA i XB i−C i YD i=E i,i=1,2,...,s.(1.8)In[31,32],Wang investigated the centro-symmetric solution to the system of quaternion matrix equationsA1X=C1,A3XB3=C3.(1.9) In[36],the maximal and minimal ranks and the least-norm of the general solution to the system A1X=C1,A2X2=C2,A3X1B1+A4X2B2=C3,(1.10) over a quaternion algebra were derived.When the solvability conditions of the system of linear real quaternion matrix equationsA1X=C1,XB1=C2,A2X2=C3,XB2=C4,A3X1B3+A4X2B4=C c,(1.11) are satisfied,Wang et al.[36]presented some necessary and sufficient conditions for the existence of a solution to this system and gave an expression of the general solution to the system.It is well-known that Sylvester and Lyapunov matrix equations are important equations which play a fundamental role in the variousfields of engineering theory,particularly in control systems. The numerical solution of Sylvester and Lyapunov matrix equations has been addressed in a large body of literature.Dehghan and Hajarian[11]proposed an efficient iterative method for solving the second-order Sylvester matrix equationEVF2−AVF−CV=BW.(1.12) Zhou and Duan[38,41,39]established the solution of the several generalized Sylvester matrix equations.Zhou et al.[40]proposed gradient-based iterative algorithms for solving the general coupled Sylvester matrix equations with weighted least squares solutions.In[42],general parametric solution to a family of generalized Sylvester matrix equations arising in linear system theory is presented by using the socalled generalized Sylvester mapping which has some elegant properties.Iterative algorithms are commonly employed to solve large linear matrix equations.By extending the well-known Jacobi and Gauss–Seidel iterations for Ax=b,Ding et al.[20]derived iterative solu-tions of the matrix equation AXB=F and the generalized Sylvester matrix equation AXB+CXD=F.In [15,16,20],to solve(coupled)matrix equations,the iterative methods are given which are based on the hierarchical identification principle[17,18].The gradient-based iterative(GI)algorithms[15,20]and least squares based iterative algorithm[16]for solving(coupled)matrix equations are innovational and computationally efficient numerical algorithms and were presented based on the hierarchical identification principle[17,18]which regards the unknown matrix as the system parameter matrix to be identified.Also Ding and Chen[19],applying the gradient search principle and the hierarchical identification principle,presented the gradient-based iterative algorithms for the general coupled matrix equationspA ij X jB ij=M i,i=1,2,...,p.(1.13)j=1We would like to comment that the coupled matrix equations(1.13)are quite general and include many matrix equations such as the generalized(coupled)Lyapunov and Sylvester matrix equations.Also by using the above-mentioned algorithms,we can not obtain the generalized bisymmetric solution group of the coupled matrix equations(1.13).In this paper,we mainly consider the following problems:Problem1.1.For given matrices A ij∈R r i×n j,B ij∈R n j×s i,M i∈R r i×s i and the symmetric orthogonal matrices R j∈SOR n j×n j,find the generalized bisymmetric matrix group(X1,X2,...,X p)with X j∈BSR n j×n j,j=1,2,...,p,such thatR jpA ij X jB ij=M i,i=1,2,...,p.j=11534M.Dehghan,M.Hajarian /Linear Algebra and its Applications 432(2010)1531–1552Problem 1.2.Let Problem 1.1be consistent,and its solution group set be denoted by S r .For a given gen-eralized bisymmetric matrix group (X 1, X 2,..., X p )with X j ∈BSR n j ×n jR j,j =1,2,...,p ,find ( X 1, X 2,..., X p )∈S r with X j ∈BSR n j ×nj R j ,j=1,2,...,p such thatp j =1|| X j − X j ||2F =min(X 1,X 2,...,X p )∈S r ⎧⎨⎩pj =1||X j − X j ||2F ⎫⎬⎭.(1.14)The rest of the paper is structured as follows.In Section 2,first we introduce an iterative method for solving Problem 1.1.Then we show that the introduced algorithm can obtain a generalized bisym-metric solution group (the minimal Frobenius norm generalized bisymmetric solution group)for any (special)initial matrix group within finite steps.Also we will solve Problem 1.2.Finally,we will give two numerical examples to verify our results in Section 4.2.Main resultsIn this section,by extending the idea of conjugate gradient method,we propose an iterative method to compute the generalized bisymmetric solution group of the matrix equations (1.13).Then some properties of such algorithm are discussed.By using these properties,we present the convergence results for the algorithm.In order to explain the method,we first give the following lemma.Lemma 2.1.The coupled matrix equations (1.13)have a generalized bisymmetric solution group if and only if the system of matrix equations⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩ p j =1A ij X j B ij =M i ,i =1,2,...,p, p j =1B T ij X j A T ij =M T i,i =1,2,...,p, pj =1A ij R j X j R j B ij =M i ,i =1,2,...,p, p j =1B T ij R j X j R j A T ij =M Ti ,i =1,2,...,p,(2.1)is consistent .Proof.If the coupled matrix equations (1.13)have a generalized bisymmetric matrix group X ∗1,X ∗2,...,X ∗p with X ∗j ∈BSR n j ×n j R j ,i.e.,X ∗j =X ∗T j =R j X ∗j R j for j =1,2,...,p ,then we getp j =1B T ij X ∗j A Tij=p j =1A ij X ∗Tj B ijT=p j =1A ij X ∗jB ijT=M T i ,(2.2)p j =1A ij R j X ∗j R j B ij=p j =1A ij X ∗jB ij=M i ,(2.3)andp j =1B T ij R j X ∗j R j A T ij=p j =1A ij R T j X ∗T j R T jB ijT=p j =1A ij X ∗j B ijT=M T i ,(2.4)for i=1,2,...,p .By combining the equalities (2.2)–(2.4),we obtain that the generalized bisymmetricmatrix group X ∗1,X ∗2,...,X ∗pis a solution group of the system of matrix equations (2.1).Conversely,suppose that the system of matrix equations (2.1)is consistent and (X 1,X 2,...,X p )isa solution group of (2.1).DefineM.Dehghan,M.Hajarian/Linear Algebra and its Applications432(2010)1531–15521535Y j=X j+R j X j R j+X T j+R j X T j R j4for j=1,2,...,p.(2.5)Obviously that Y j∈BSR n j×n jR jfor j=1,2,...,p.Now we can write pj=1A ij Y j B ij=pj=1A ijX j+R j X j R j+X T j+R j X T j R j4B ij=14⎡⎣pj=1A ij X jB ij+pj=1A ij R j X j R jB ij+pj=1A ij X T jB ij+pj=1A ij R j X T j R jB ij⎤⎦=14⎡⎣pj=1A ij X jB ij+pj=1A ij R j X j R jB ij+pj=1B T ij X j A T ijT+pj=1B T ij R j X j R j A T ijT⎤⎦=14M i+M i+(M T i)T+(M T i)T=M i,(2.6)for i=1,2,...,p.Hence(Y1,Y2,...,Y p)is a solution group of the coupled matrix equations(1.13). The proof is completed.By considering Lemma2.1,the solvability of the system of matrix equations(2.1)is equivalent to Problem1.1.Also the system of matrix equations(2.1)is equivalent to the following system:⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝B T11⊗A11B T12⊗A12...B T1p⊗A1p............B T p1⊗A p1B T p2⊗A p2...B T pp⊗A ppA11⊗B T11A12⊗B T12...A1p⊗B T1p............A p1⊗B T p1A p2⊗B T p2...A pp⊗B T ppB T11R1⊗A11R1B T12R2⊗A12R2...B T1p R p⊗A1p R p............B T p1R1⊗A p1R1B T p2R2⊗A p2R2...B T pp R p⊗A pp R pA11R1⊗B T11R1A12R2⊗B T12R2...A1p R p⊗B T1p R p............A p1R1⊗B T p1R1A p2R2⊗B T p2R2...A pp R p⊗B T pp R p⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝vec(X1)vec(X2).................vec(X p)⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝vec(M1)...vec(M p)vec(M T1)...vec(M T p)vec(M1)...vec(M p)vec(M T1)...vec(M T p)⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠.(2.7)The size of the system of the linear equations(2.7)is large.However,iterative methods consume more computer time and memory once the size of the system is large.To overcome these complications and drawbacks,by extending the idea of conjugate gradient method,we propose an iterative method for solving(1.13).Consider the linear algebraic system of equationsAx=b,(2.8) where we have denoted by A=(a ij)∈R n×n the coefficient matrix,by b=(b i)∈R n the right side vector and by x=(x i)∈R n the unknown vector,respectively.For solving systems of linear algebraic equations,the conjugate gradient method reads as follows:1536M.Dehghan,M.Hajarian/Linear Algebra and its Applications432(2010)1531–1552Conjugate gradient algorithm1.Select x(0)∈R n and set r(0)=b−Ax(0).Setα(0)=||r(0)||2and d(0)=r(0);2.for k=0,1,...until convergence do:3.s(k)=Ad(k);4.t(k)=α(k)/(d(k)T s(k));x(k+1)=x(k)+t(k)d(k);r(k+1)=r(k)−t(k)s(k);β(k+1)=||r(k+1)||2/||r(k)||2;d(k+1)=r(k+1)+β(k+1)d(k);5.end for.If A be a symmetric and positive definite matrix,then any method which employs conjugate direc-tions to solve(2.8)terminates after at most n steps,yielding the exact solution[21,25].In general,CG method cannot guarantee that x(k)converges to the exact solution x=A−1b.Also this method not suitable for solving the non-square system:Bx=c with B∈R m×n.This motivates us to study new iterative methods.It is obvious that CG method can be represented asx(k+1)=x(k)+t(k)d(k),(2.9) where parameter t(k)and vector d(k)to be determined.Now we decompose the general coupled matrix equations(1.13)into p subsystems and consider a large family of CG iterative methods for the subsystems as:X i(k+1)=X i(k)+T i(k)D i(k)for i=1,2,...,p,(2.10) where parameter T i(k)and matrix D i(k)(for i=1,2,...,p)to be determined.According to(2.10),we propose an iterative algorithm for solving Problem1.1.The details of algorithm are given as follows. Algorithm1Step1.Input matrices A ij∈R r i×n j,B ij∈R n j×s i,M i∈R r i×s i and the symmetric orthogonal matrices R j∈SOR n j×n j;Step2.Choose arbitrary X j(1)∈BSR n j×n jR jfor j=1,2,...,p;Step3.CalculateR(1)=diagM1−pt=1A1t X t(1)B1t,M2−pt=1A2t X t(1)B2t,...,M p−pt=1A pt X t(1)B pt;P i(1)=14⎧⎨⎩ps=1A T siM s−pt=1A st X t(1)B stB T si+ps=1⎡⎣BsiM s−pt=1A st X t(1)B stTA si⎤⎦+ps=1R i A T siM s−pt=1A st X t(1)B stB T si R i+ps=1⎡⎣Ri B siM s−pt=1A st X t(1)B stTA si R i⎤⎦⎫⎬⎭,i=1,2,...,p;k:=1;Step4.If R(k)=0,then stop and(X1(k),X2(k),...,X p(k))is the generalized bisymmetric solution group;else if R(k)/=0but P m(k)=0for all m=1,2,...,p,then stop and the general coupled matrix equations(1.13)are not consistent over generalized bisymmetric matrix group;else k:=k+1;Step5.CalculateX i(k)=X i(k−1)+||R(k−1)||2Fpt=1P t k1FP i(k−1),i=1,2,...,p;M.Dehghan,M.Hajarian /Linear Algebra and its Applications 432(2010)1531–15521537R (k )=diagM 1−p t =1A 1t X t (k )B 1t ,M 2−p t =1A 2t X t (k )B 2t ,...,M p −p t =1A pt X t (k )B pt;=R (k −1)−||R (k −1)||2Fpt =1||P t (k −1)||2F×diagpt =1A 1t P t (k −1)B 1t ,p t =1A 2t P t (k −1)B 2t ,...,p t =1A pt P t (k −1)B pt;P i (k )=14⎡⎣p s =1A T si M s−p t =1A st X t (k )B stB Tsi+p s =1B si M s−p t =1A st X t (k )B stTA si +p s =1R i A T si M s−p t =1A st X t (k )B stB Tsi R i+p s =1R i B siM s−p t =1A st X t (k )B stTA si R i ⎤⎦+||R (k )||2F||R (k −1)||2FP i (k −1),i =1,2,...,p ;Step 6.Go to Step 4.Remark 2.1.X j (k )and P j (k )(j =1,2,...,p )generated by Algorithm 1are the generalized bisymmetricmatrices with respect to R j ,i.e.,X j (k )=X T j (k )=R j X j (k )R j and P j (k )=P T j (k )=R j P j (k )R j ,for k =1,2,...,j =1,2,...,p .(2.11)Remark 2.2.Because of the influence of the error of calculation,the residual R (k )(k =1,2,...)is usually unequal to zero exactly in the process of the iteration.We regard the matrix R (k )as a zero matrix if ||R (k )||F <εwhere εis a small positive number.In Algorithm 1,the iteration will be stopped whenever ||R (k )||F <ε.We begin with the following useful lemmas about Algorithm 1to be used in the next results.Lemma 2.2.Suppose that the sequences {R (k )}and {P i (k )}for i =1,2,...,p and k =1,2,...,s (R (k )/=0,k =1,2,...,s )are generated by Algorithm 1,thentrR T(m )R (n )=0andp i =1trP T i (m )P i (n )=0,for m,n =1,2,...,s (m /=n ).(2.12)The proof of Lemma 2.2is presented in the Appendix.Lemma 2.3.Assume that the coupled matrix equations (1.13)are consistent over generalized bisymmetric matrix group (X 1,X 2,...,X p )and alsoX ∗1,X ∗2,...,X ∗pis an arbitrary generalized bisymmetric solution group of (1.13),then for any initial matrix group (X 1(1),X 2(1),...,X p (1))with X j (1)∈BSR nj ×n jR jandj=1,2,...,p,the sequences {X i (k )},{R (k )}and {P i (k )}(i =1,2,...,p )generated by Algorithm 1satisfyp i =1tr X ∗i −X i (n )T P i (m )=0for m <n,(2.13)1538M.Dehghan,M.Hajarian /Linear Algebra and its Applications 432(2010)1531–1552andp i =1trX ∗i−X i (n ) TP i (m )=||R (m )||2F for m n .(2.14)The proof of Lemma 2.3is given in the Appendix.Remark 2.3.If there exists a positive number j such that P i (j )=0for all i =1,2,...,p but R (j )/=0,then by considering Lemma 2.3,we can get the coupled matrix equations (1.13)are not consistent over generalized bisymmetric matrices.Hence,the solvability of Problem 1.1can be determined by Algorithm 1in the absence of roundoff errors.Theorem 2.1.Suppose that Problem 1.1is consistent .Then for any arbitrary initial matrix group (X 1(1),X 2(1),...,X p (1))with X j (1)∈BSR n j ×n jR jand j =1,2,...,p,a generalized bisymmetric solution group of Problem 1.1can be obtained with finite iteration steps in the absence of roundoff errors .Proof.Let R (i )/=0for i =1,2,...,m = pi =1r i s i ,then from Lemma 2.3and Remark 2.3we haveP g (i )/=0for some g ∈{1,2,...,p }.Hence R m +1andX (m +1)1,X (m +1)2,...,X (m +1)pcan be computed.Now from Lemma 2.2,it is not difficult to gettr (R T (m +1)R (i ))=0i =1,2,...,m,(2.15)andtr (R T (i )R (j ))=0i,j =1,2,...,m,i /=j .(2.16)Then R (1),R (2),...,R (m )is an orthogonal basis of the subspaceK = M |M =diag Z 1,Z 2,...,Z p where Z i ∈R r i ×s ifor i =1,2,...,p .It follows that R (m +1)=0and (X 1(m +1),X 2(m +1),...,X p (m +1))is a solution group of Prob-lem 1.1.Therefore when Problem 1.1is consistent,we can verify that the solution group of Problem 1.1can be obtained within finite iterative steps. Let E j be arbitrary matrices for j=1,2,...,p ,we can get⎛⎜⎜⎜⎜⎜⎜⎜⎝vecpj =1A T j 1E j B Tj 1+pj =1B j 1E T j A j 1+ p j =1R 1A T j 1E j B T j 1R 1+ p j =1R 1B j 1E Tj A j 1R 1vec p j =1A T j 2E j B T j 2+ p j =1B j 2E T j A j 2+ p j =1R 2A T j 2E j B T j 2R 2+ p j =1R 2B j 2E T j A j 2R 2 .........vec p j =1A T jp E j B T jp + p j =1B jp E T j A jp + p j =1R p A T jp E j B T jp R p + p j =1R p B jp E T j A jp R p⎞⎟⎟⎟⎟⎟⎟⎟⎠=⎛⎜⎜⎜⎜⎜⎜⎝B 11⊗A T 11...B p 1⊗A T p 1A T 11⊗B 11...A T p 1⊗B p 1R 1B 11⊗R 1A T 11B 12⊗A T 12...B p 2⊗A T p 2A T 12⊗B 12...A T p 2⊗B p 2R 2B 12⊗R 2A T 12.....................B 1p⊗A T 1p...B pp ⊗A T pp A T 1p ⊗B 1p...A T pp ⊗B pp R p B 1p ⊗R p A T 1p...R 1B p 1⊗R 1A T p 1R 1A T 11⊗R 1B 11...R 1A T p 1⊗R 1B p 1...R 2B p 2⊗R 2A T p 2R 2A T 12⊗R 2B 12...R 2A T p 2⊗R 2B p 2..................R p B pp ⊗R p A T pp R p A T 1p ⊗R p B 1p...R p A Tpp ⊗R p B pp⎞⎟⎟⎟⎟⎟⎟⎠M.Dehghan,M.Hajarian/Linear Algebra and its Applications432(2010)1531–15521539×⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝vec(E1)...vec(E p)vec(E T1)...vec(E T p)vec(E1)...vec(E p)vec(E T1)...vec(E T p)⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝B T11⊗A11B T12⊗A12...B T1p⊗A1p............B T p1⊗A p1B T p2⊗A p2...B T pp⊗A ppA11⊗B T11A12⊗B T12...A1p⊗B T1p............A p1⊗B T p1A p2⊗B T p2...A pp⊗B T ppB T11R1⊗A11R1B T12R2⊗A12R2...B T1p R p⊗A1p R p............B T p1R1⊗A p1R1B T p2R2⊗A p2R2...B T pp R p⊗A pp R pA11R1⊗B T11R1A12R2⊗B T12R2...A1p R p⊗B T1p R p............A p1R1⊗B T p1R1A p2R2⊗B T p2R2...A pp R p⊗B T pp R p⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠T⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝vec(E1)...vec(E p)vec(E T1)...vec(E T p)vec(E1)...vec(E p)vec(E T1)...vec(E T p)⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠∈R ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝B T11⊗A11B T12⊗A12...B T1p⊗A1p............B T p1⊗A p1B T p2⊗A p2...B T pp⊗A ppA11⊗B T11A12⊗B T12...A1p⊗B T1p............A p1⊗B T p1A p2⊗B T p2...A pp⊗B T ppB T11R1⊗A11R1B T12R2⊗A12R2...B T1p R p⊗A1p R p............B T p1R1⊗A p1R1B T p2R2⊗A p2R2...B T pp R p⊗A pp R pA11R1⊗B T11R1A12R2⊗B T12R2...A1p R p⊗B T1p R p............A p1R1⊗B T p1R1A p2R2⊗B T p2R2...A pp R p⊗B T pp R p⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠T⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠.(2.17)Now if we consider the initial matricesX i(1)=pj=1A T ji E jB T ji+pj=1B ji E T j A ji+pj=1R i A T ji E j B T ji R i+pj=1R i B ji E T j A ji R i,i=1,2,...,p,(2.18)then all X i(k)for i=1,2,...,p generated by Algorithm1satisfy⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝vec(X1(k))vec(X2(k))vec(X3(k))..................vec(X p(k))⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠∈R⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝B T11⊗A11B T12⊗A12...B T1p⊗A1p............B T p1⊗A p1B T p2⊗A p2...B T pp⊗A ppA11⊗B T11A12⊗B T12...A1p⊗B T1p............A p1⊗B T p1A p2⊗B T p2...A pp⊗B T ppB T11R1⊗A11R1B T12R2⊗A12R2...B T1p R p⊗A1p R p............B T p1R1⊗A p1R1B T p2R2⊗A p2R2...B T pp R p⊗A pp R pA11R1⊗B T11R1A12R2⊗B T12R2...A1p R p⊗B T1p R p............A p1R1⊗B T p1R1A p2R2⊗B T p2R2...A pp R p⊗B T pp R p⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠T⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠.1540M.Dehghan,M.Hajarian /Linear Algebra and its Applications 432(2010)1531–1552By using the Moore–Penrose pseudoinverse to solve a linear systems Ax =b ,it is well known that for a consistent linear system Ax =b ,all the solutions are given by x =A †b +Null (A );hence A †b gives the minimum norm solution.From Range (A †)=R (A T ),we have y ∗∈R (A T )is a least 2-norm solution of the system of linear equations.Hence with the initial matrix group (X 1(1),X 2(1),...,X p (1))whereX i (1)=p j =1A T ji E jB T ji +p j =1B ji E T j A ji +p j =1R i A T ji E j B Tji R i +p j =1R i B ji E T j A ji R i ,i=1,2,...,p,it follows that the generalized bisymmetric solution group obtained by Algorithm 1is the least Frobe-nius norm generalized bisymmetric solution group.By using the above conclusions,we can presentthe following theorem.Theorem 2.2.Suppose that Problem 1.1is consistent .Let the initial iteration matrices beX i (1)=p j =1A T ji E jB T ji +p j =1B ji E T j A ji +p j =1R i A T ji E j B Tji R i +p j =1R i B ji E T j A ji R i ,i=1,2,...,p,where E j be arbitrary matrices,or especially,X i (1)=0for j =1,2,...,p,then the solution group X ∗1,X ∗2,...,X ∗p,generated by Algorithm 1,is the least Frobenius norm generalized bisymmetric solution group of the coupled matrix equations (1.13).Now we study Problem 1.2as follows.When Problem 1.1is consistent,its solution group set S r is nonempty.For a given matrix group( X 1, X 2,..., X p )with X j ∈BSR n j ×n jR j,j =1,2,...,p ,it is not difficult to getp j =1A ij X jB ij=M i ,i =1,2,...,p ⇔p j =1A ij (X j − X j )B ij =M i −p j =1A ij X jB ij ,i=1,2,...,p .Define X j =X j − X j and M i =M i − p j =1A ij X j B ij for i =1,2,...,p .Then the matrix nearness Problem 1.2is equivalent to first finding the least Frobenius norm generalized bisymmetric solution group X ∗1,X ∗2,...,X ∗p of the new coupled matrix equationsp j =1A ij X jB ij=M i ,i =1,2,...,p,(2.19)which can be obtained by using Algorithm 1with the initial generalized bisymmetric matricesX i (1)=p j =1A T ji E jB Tji +p j =1B ji E T j A ji +p j =1R i A T ji E j B Tji R i +p j =1R i B ji E T j A ji R i ,i=1,2,...,p,where E j be arbitrary matrices,or especially,X i (1)=0for j =1,2,...,p .Here the generalized bisym-metric solution group of the matrix nearness Problem 1.2can be represented as ( X 1, X 2,..., X p )withX j=X ∗j +X j ,j =1,2,...,p .(2.20)3.Numerical experimentsIn this section,we reported two numerical examples to compute the generalized bisymmetric solution group of coupled matrix equations.We implemented the algorithms in MATLAB and run the programs on a Pentium IV.Meanwhile,we assumed that ε=10−12.。