Research Article An Iterative Algorithm for the Reflexive Solution of the General Coupled Matrix Equations

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1 he Scientific World Journal Volume 013 Article ID ages htt://dxdoiorg/101155/013/95974 Research Article An Iterative Algorithm for the Reflexive Solution of the General Couled Matrix Euations Zhongli Zhou and Guangxin Huang Geomathematics Key Laboratory of Sichuan Province College of Management Science Chengdu University of echnology Chengdu China Corresondence should be addressed to Guangxin Huang; Received 3 August 013; Acceted 6 August 013 Academic Editors: R Camoamor-Stursberg I Ivanov and Y Shi Coyright 013 Z Zhou and G Huang his is an oen access article distributed under the Creative Commons Attribution License which ermits unrestricted use distribution and reroduction in any medium rovided the original work is roerly cited he general couled matrix euations including the generalized couled Sylvester matrix euations as secial cases have numerous alications in control and system theory In this aer an iterative algorithm is constructed to solve the general couled matrix euations over reflexive matrix solution When the general couled matrix euations are consistent over reflexive matrices the reflexive solution can be determined automatically by the iterative algorithm within finite iterative stes in the absence of round-off errors he least robenius norm reflexive solution of the general couled matrix euations can be derived when an aroriate initial matrix is chosen urthermore the uniue otimal aroximation reflexive solution to a given matrix grou in robenius norm can be derived by finding the least-norm reflexive solution of the corresonding general couled matrix euations A numerical examle is given to illustrate the effectiveness of the roosed iterative algorithm 1 Introduction Let P R n n be a generalized reflection matrix; that is P Pand P IAmatrixA R n n is called reflexive withresecttothematrixp if PAP Ahesetofalln-by-n reflexive matrices with resect to the generalized reflection matrix P is denoted by R n n r P LetR m n denote the set of all m nreal matrices We denote by the suerscrit the transose of a matrix In matrix sace R m n define inner roduct as; A B trb A for all A B R m n ; A reresents the robenius norm of A RA reresents the column sace ofavec reresents the vector oerator; that is veca a 1 a a n R mn for the matrix A a 1 a a n R m n a i R m i 1n A BstandsfortheKroneckerroductofmatricesA and B In this aer we will consider the following two roblems Problem 1 Let P j R n j n j be generalized reflection matrices or given matrices A ij R r i n j B ij R n j s i andm i R r i s i find reflexive matrix solution grou X 1 X X with X j R n j n j r P j such that A ij X j B ij M i 1 Problem When Problem 1 is consistent let S E denote the set of the reflexive solution grou of Problem 1;thatis S E { {X 1 X X { A ij X j B ij M i X j R n j n j r P j } } } or a given reflexive matrix grou X 0 1 X0 X0 Rn 1 n 1 r P 1 R n n r P R n n r P 3

2 he Scientific World Journal ind X 1 X X S E such that X j Xj 0 { { X 1 X X S E min { X j Xj 0 } } } he general couled matrix euations 1 including the generalized couled Sylvester matrix euations as secial cases may arise in many areas of control and system theory Many theoretical and numerical results on 1 and some of its secial cases have been obtained Least-suares-based iterative algorithms are very imortant in system identification arameter estimation and signal rocessing including the recursive least suares RLS and iterative least suares ILS methods for solving the solutions of some matrix euations for examle the Lyaunov matrix euation Sylvester matrix euations and couled matrix euations as well or examle novel gradient-based iterative GI method [1 5] and leastsuares-based iterative methods [3 4 6] withhighlycomutational efficiencies for solving couled matrix euations areresentedandhavegoodstabilityerformancesbased on the hierarchical identification rincile which regards the unknown matrix as the system arameter matrix to be identified Ding and Chen [1] resented the gradient-based iterative algorithms by alying the gradient search rincile and the hierarchical identification rincile for 1with Wu et al [7 8] gavethefiniteiterativesolutionstocouled Sylvester-conjugate matrix euations Wu et al [9] gave the finite iterative solutions to a class of comlex matrix euations withconjugateandtransoseoftheunknownsjonsson and Kågström [10 11] roosed recursive block algorithms for solving the couled Sylvester matrix euations and the generalizedsylvesterandlyaunovmatrixeuationsby extending the idea of conjugate gradient method Dehghan and Hajarian [1] constructed an iterative algorithm to solve 1 with over generalized bisymmetric matrices Very recently Huang et al [13] resented a finite iterative algorithms for the one-sided and generalized couled Sylvester matrix euations over generalized reflexive solutions Yin et al [14] resentedafiniteiterativealgorithmsforthetwosided and generalized couled Sylvester matrix euations over reflexive solutions or more results we refer to [15 8] However to our knowledge the reflexive solution to the general couled matrix euations 1 and the otimal aroximation reflexive solution have not been derived In this aer we will consider the reflexive solution of 1 and the otimal aroximation reflexive solution hisaerisorganizedasfollowsinsection wewill solve Problem 1 by constructing an iterative algorithm he convergence of the roosed algorithm is roved or any arbitrary initial matrix grou we can obtain a reflexive solution grou of Problem 1 within finite iteration stes in the absence of round-off errors urthermore for a secial initialmatrixgrouwecanobtaintheleastrobeniusnorm solution of Problem 1hen in Section 3we give the otimal aroximate solution grou of Problem by finding the 4 least robenius norm reflexive solution grou of the corresonding general couled matrix euations In Section 4 a numerical examle is given to illustrate the effectiveness of our method At last some conclusions are drawn in Section 5 An Iterative Algorithm for Solving Problem 1 In this section we will first introduce an iterative algorithm to solve Problem 1 then rove its convergence We will also give the least-norm reflexive solution of Problem 1 when an aroriate initial iterative matrix grou is chosen Algorithm 3 Ste 1InutmatricesA ij R r i n j B ij R n j s i M i R r i s i and generalized reflection matrices P j R n j n j i 1 Ste Choose an arbitrary matrix grou Comute X 1 1 X 1 X 1 R n 1 n 1 r P 1 R n n r R 1 diag M 1 S j 1 1 [ A ij M i P R n n r P A 1l X l 1 B 1l M A l X l 1 B l M P j A ij M i A il X l 1 B il B ij 5 A l X l 1 B l A il X l 1 B il B ij P j] k:1 6 Ste 3 IfRk 0 thenstoandx 1 k X kx k is the solution grou of 1; elseif Rk 0butS j k 0 thenstoand1 are not consistent over reflexive matrix grou; else k:k1 Ste 4Comute X j k X j k 1 S j k 1 R k diag M 1 R k 1 S l k 1 A 1l X l k B 1l M A l X l k B l M A l X l k B l

3 he Scientific World Journal 3 Rk 1 diag R k 1 S l k 1 A 1l S l k 1 B 1l S j k 1 [ A ij M i Ste 5GotoSte3 A l S l k 1 B l P j A ij M i A il X l k B il B ij R k R k 1 S j k 1 A l S l k 1 B l A il X l k B il B ij P j] ObviouslyitcanbeseenthatX j k S j k R n j n j r P j for all and k1 Lemma 4 or the seuences {Rk} {S j k} j 1 generated byalgorithm 3and m we have tr R s R t 0 tr S j s S j t 0 st1mst he roof of Lemma 4 is resented in the aendix Lemma 5 Suose that X 1 X X is an arbitrary reflexive solution grou of Problem 1; thenforanyinitial reflexive matrix grou X 1 1 X 1X 1onehas tr X j X j k S j k R k k1 where the seuences {X j k} {S j k}and{rk} are generated by Algorithm 3 he roof of Lemma 5 is resented in the aendix Remark 6 If there exists a ositive number k such that S j k 0 j 1but Rk 0thenbyLemma 5 we get that 1 are not consistent over reflexive matrices heorem 7 Suose that Problem 1 is consistent; then for an arbitrary initial matrix grou X 1 X X with X j R n j n j r P j a reflexive solution grou of Problem 1 can be obtained with finite iteration stes in the absence of round-off errors Proof If Rk 0 k 1m r is i thenby Lemma 5 and Remark 6 we have S j k 0 for all j 1and k1mhuswecancomuterm 1 and X 1 m 1 X m1x m 1 by Algorithm 3 By Lemma 4wehave tr R m1 R k0 k1m tr R k R l0 km kl 10 ItcanbeseenthatthesetofR1 R Rm is an orthogonal basis of the matrix subsace S{L Ldiag L 1 L L L i R r i s i } 11 which imlies that Rm 1 0;thatisX 1 m 1 X m 1X m 1 with X j m 1 R n j n j r P j is a reflexive solution grou of Problem 1 hiscomletesthe roof o show the least robenius norm reflexive solution of Problem 1 we first introduce the following result Lemma 8 see [0Lemma4] Suosethattheconsistent system of linear euation Axbhas a solution x RA ; then x is a uniue least robenius norm solution of the system of linear euation By Lemma 8 the following result can be obtained heorem 9 Suose that Problem 1 is consistent If one chooses the initial iterative matrices X j 1 A ij K ib ij P ja ij K ib ij P j j 1 where K i R r i s i i 1 are arbitrary matrices esecially X j 1 0 R n j n j P j then the solution grou X 1 X X generated by Algorithm 3 is the uniue least robenius norm reflexive solution grou of Problem 1 Proof We know that the solvability of 1 over reflexive matrices is euivalent to the following matrix euations: A ij X j B ij M i A ij P j X j P j B ij M i 1

4 4 he Scientific World Journal hen the system of matrix euations 1iseuivalentto B 11 A 11 B 1 A 1 B 1 A 1 B A B 11 P 1 A 11 P 1 B 1 P A 1 P B 1 P 1 A 1 P 1 B P A P vec X 1 vec X vec M 1 vec M vec M 1 vec M 13 Let X j 1 A ij K ib ij P ja ij K ib ij P j where K i R r i s i are arbitrary matrices; then vec X 1 1 vec X 1 vec vec A i1 K ib i1 P 1 A i1 K ib i1 P 1 A i K ib i P A i K ib i P vec K 1 B 11 A 11 B 1 A 1 P 1B 11 P 1 A 11 P 1B 1 P 1 A 1 vec K B 1 A 1 B A P B 1 P A 1 P vec K B P A 1 vec K B 11 A 11 B 1 A 1 B 1 A 1 B A B 11 P 1 A 11 P 1 B 1 P A 1 P B 1 P 1 A 1 P 1 B P A P vec K 1 vec K vec K 1 vec K B 11 A 11 B 1 A 1 R B 1 A 1 B A B 11 P 1 A 11 P 1 B 1 P A 1 P B 1 P 1 A 1 P 1 B P A P 14

5 he Scientific World Journal 5 urthermore we can see that all reflexive matrix solution grous X 1 k X kx k generated by Algorithm 3 satisfy vec X 1 1 vec X 1 By using Algorithm 3 let initial iteration matrices X j 1 A ij K ib ij P j A ij K ib ij P j 18 B 11 A 11 B 1 A 1 R B 1 A 1 B A B 11 P 1 A 11 P 1 B 1 P A 1 P B 1 P 1 A 1 P 1 B P A P ; 15 by Lemma 8 we know that X 1 X X is the least robenius norm reflexive solution grou of the system of linear euation 13 Since vector oerator is isomorhic X 1 X X is the uniue least robenius norm reflexive solution grou of the system of matrix euations 1 hus X 1 X X is the uniue least robenius norm reflexive solution grou of Problem 1hiscomletestheroof 3 he Solution of Problem In this section we will show that the reflexive solution grou of Problem to a given reflexive matrix grou can be derived by finding the least robenius norm reflexive solution grou of the corresonding general couled matrix euations When Problem 1 is consistent the set of the reflexive solution grous of Problem 1 denoted by S E is not emty or a given matrix air X 0 1 X0 X0 with X0 j Rn j n j r P j wehave A ij X j B ij M i M i A ij X j X 0 j B ij A ij X 0 j B ij 16 Set X j X j X 0 j and M i M i A ijx 0 j B ij; then solving Problem is euivalent to finding the least robenius norm reflexive solution grou X 1 X X of the corresonding general couled matrix euations A ij X j B ij M i 17 where K i R r i s i i 1 are arbitrary matrices esecially X j 1 0 R n j n j P j j 1;thenwe can get the the least robenius norm reflexive solution grou X 1 X X of 17 hus the reflexive solution grou of Problem canbereresentedas X 1 X X X 1 X0 1 X X0 X X A Numerical Examle In this section we will show a numerical examle to illustrate our results All the tests are erformed by MALAB 78 Examle 10 Consider the reflexive solution of the general couled matrix euations A 11 X 1 B 11 A 1 X B 1 M 1 A 1 X 1 B 1 A X B M 0 where A B A B

6 6 he Scientific World Journal Let A B A B M M P P be the generalized reflection matrices We will find the reflexive solution of the the general couled matrix euations 0 by using Algorithm 3 It can be verified that the matrix euations 0areconsistentover reflexive matrices and the solution is X X Because of the influence of the error of calculation the residual Rk is usually uneual to zero in the rocess of the iteration where k 1 oranychosenositive number ε however small enough for examle ε 10000e 010 whenever Rk < ε stotheiteration; X 1 k X k is regarded to be the reflexive solution of the matrix euations 0 Choose an initially iterative matrix grou X 1 1 X 1suchas X X 1 ; by Algorithm 3wehave X 1 X X X R e 011 < ε 7 So we obtain the reflexive solution of the matrix euations 0 he relative error of the solution and the residual are shown in igure 1wheretherelativeerrorREk X 1 k X 1 X k X / X 1 X and the residual Rk Rk Let S E denote the set of all reflexive solution grou of the matrixeuations 0 or two given reflexive matrices X X we will find X 1 X S E suchthat 8 X 1 X1 0 X X 0 min X 1 X S E X 1 X1 0 X X1 0 ; 9 that is find the otimal aroximate reflexive solution grou to the given matrix grou X 0 1 X0 in S E in robenius norm

7 he Scientific World Journal 7 Let X 1 X 1 X 0 1 X X X 0 M 1 M 1 A 11 X 0 1 B 11 A 1 X 0 B 1 M M A 1 X 0 1 B 1 A X 0 B by the method mentioned in Section 3 we can obtain the least-norm reflexive solution grou X 1 X of the matrix euations A 11 X 1 B 11 A 1 X B 1 M 1 and A 1 X 1 B 1 A X B M by choosing the initially iterative matrices X and X 1 0;thenbyAlgorithm 3 we have that X 1 X X X R e 011 < ε 10000e Relative error/residual REk 0001 Rk k iteration ste igure 1: he relative error of the solutions and the residual for Examle 10 with X and X 1 0 and the otimal aroximate reflexive solution to the matrix grou X 0 1 X0 in robenius norm are X 1 X 1 X X X X he relative error and the residual of the solution are shown in igure wheretherelativeerrorrek X 1 k X 0 1 X 1 X k X 0 X / X 1 X and the residual Rk Rk 5 Conclusions In this aer an iterative algorithm is resented to solve the general couled matrix euations A ijx j B ij M i i 1over reflexive matrices When the general couled matrix euations are consistent over reflexive matrices for any initially reflexive matrix grou the reflexive solution grou can be obtained by the iterative algorithm within finite iterative stes in the absence of round-off errors When a secial kind of initial iteration matrix grou is given the uniue least-norm reflexive solution of the general couled matrix euations can be derived urthermore the otimal aroximate reflexive solution of the general couled Relative error/residual REk 0001 Rk k iteration ste igure : he relative error of the solutions and the residual for Examle 10 with X 0 1 and X0 matrix euations to a given reflexive matrix grou can be derived by finding the least-norm reflexive solution of new corresonding general couled matrix euations inally a numerical examle is given in Section4 to illustrate that our iterative algorithm is uite effective

8 8 he Scientific World Journal Aendices A he Proof of Lemma 4 Since trrs Rt trrt Rs and trs j s S j t trs j t S j s for all st 1m and we only need to rove that tr R s R t 0 tr S j s S j t 0 1 t<s m A1 We rove the conclusion by induction and two stes are reuired Ste 1wewillshowthat M R 1 R 1 S j 1 tr diag A l X l 1 B l A 1j S j 1 B 1j M 1 A 1l X l 1 B 1l A j S j 1 B j tr R k1 R k 0 tr S j k1 S j k 0 k1m 1 A M A l X l 1 B l A j S j 1 B j o rove this conclusion we also use induction or k1byalgorithm 3wehavethat tr R R 1 tr [ R 1 [ diag R 1 S j 1 A 1j S j 1 B 1j R 1 R 1 S j 1 tr [ diag [ diag M 1 A j S j 1 B j A j S j 1 B j ] R 1 ] A 1j S j 1 B 1j A j S j 1 B j A 1l X l 1 B 1l M A l X l 1 B l A j S j 1 B j ] ] M R 1 R 1 S j 1 tr M i B ij S j 1 A ij R 1 R 1 S j 1 tr S j 1 [ R 1 R 1 S j 1 tr S j 1 A l X l 1 B l A il X l 1 B il A ij M i A il X l 1 B il B ij ] [ A ij M i A ilx l 1 B il B ij P ja ij M i A ilx l 1 B il B ij P j ]

9 he Scientific World Journal 9 R 1 R 1 S j 1 tr S j 1 S j 1 0 tr S j S j 1 tr [ 1 A ij M i tr P j A ij M i B ij Q j A il X l B il B ij R R 1 S j 1] S j 1 A ij M i A il X l B il A il X l B il B ij A j S j 1 B j R R 1 S j 1 S j 1 R 1 tr R R 1 R R R 1 S j 1 S j 1 R 1 tr R R 1 0 Assume that Aholds forkm 1;that is When kmwehavethat tr R m1 R m tr R m R m 1 0 tr S j m S j m 1 0 A3 A4 R R 1 S j 1 S j 1 tr S j 1 A ij M i R R 1 tr S j 1 S j 1 tr M i R R 1 S j 1 tr diag M 1 diag M M A il X l B il A il X l B il B ij A 1l X l B 1l A ij S j 1 B ij A l X l B l A l X l B l A 1j S j 1 B 1j A j S j 1 B j tr [ R m [ diag R m R m S j m tr [ diag [ diag M 1 R m S j m A 1j S j m B 1j A j S j m B j A j S j m B j ] R m ] A 1j S j m B 1j A j S j m B j ] ] A j S j m B j A 1l X l m B 1l M A l X l m B l M A l X l m B l

10 10 he Scientific World Journal R m R m S j m tr diag A 1j S j m B 1j M 1 M M R m R m S j m tr A 1l X l m B 1l A j S j m B j A l X l m B l A j S j m B j B ij S j m A ij M i R m R m S j m tr [ S j m A ij M i A l X l m B l A il X l m B il R m R m S j m tr S j m A il X l m B il B ij ] [ A ij M i A ilx l m B il B ij P ja ij M i A ilx l m B il B ij P j ] R m R m S j m tr S j m S j m R m R m 1 S j m 1 0 A5 tr S j m1 S j m tr [ 1 A ij M i tr B ij P j P j A ij M i A il X l m1 B il B ij A il X l m1 B il R m1 R m S j m ] S j m A ij M i A il X l m1 B il B ij R m1 R m S j m S m j trs j m A ij M i R m1 R m tr M i R m1 R m tr diag M 1 diag M tr S j m S j m A il X l m1 B il S j m M A il X l m1 B il B ij A 1l X l m1 B 1l A 1j S j m B 1j A ij S j m B ij A l X l m1 B l A l X l m1 B l A j S j m B j

11 he Scientific World Journal 11 R m1 R m A j S j m B j S j m R m S j m tr R m1 R m Rm1 R m1 R m S j m S j m R m tr R m1 R m 0 A6 Hence A holdsforkm herefore A holdsbythe rincile of induction Ste We show that When k1a7holds Assume that tr R k R t 0 then we show that In fact we have that tr R k1 R t tr [ R k [ tr R k1 R t 0 tr S j k1 S j t 0 t 1 k k 1 tr R k1 R t 0 tr S j k S j t 0 t1k 1 k ; tr S j k1 S j t 0 R k S j k t1k A7 A8 A9 diag tr R k R t tr [ [ diag diag M 1 M A 1j S j k B 1j A j S j k B j A j S j k B j ] R t ] R k S j k A 1j S j k B 1j A j S j k B j M R k S j k tr diag M 1 M A 1l X l t B 1l A l X l t B l A l X l t B l A 1j S j k B 1j A 1l X l t B 1l A j S j k B j A j S j k B j M R k S j k A l X l t B l A l X l t B l A j S j k B j ] ]

12 1 he Scientific World Journal tr R k S j k tr B ij S j k A ij M i S j k [ R k S j k tr S j k A il X l t B il A ij M i A il X l t B il B ij ] [ A ij M i A ilx l t B il B ij P ja ij M i A ilx l t B il B ij P j ] R k S j k tr S j k S j t R t R t 1 S j t 1 R k tr S S j k S j t j k tr P j A ij M i R k1 R k S j k ] S j t A ij M i A il X l k1 B il B ij P j A il X l k1 B il B ij R k1 R k S j k S j t tr S j t R k1 R k tr M i tr diag M 1 M diag M A ij M i tr S j k S j t A il X l k1 B il A il X l k1 B il B ij A 1l X l k1 B 1l A l X l k1 B l A ij S j t B ij A l X l k1 B l A 1j S j t B 1j R k R t S j k R t 1 tr S j k S j t 1 0 A10 S j t R t A j S j t B j tr R k1 R t Rt1 A j S j t B j rom the above results we have trrk 1 Rt 1 0 t1k 1and tr S j k1 S j t tr [ 1 A ij M i A il X l k1 B il B ij S j t R t tr R k1 R t 0 A11 By the rincile of induction A7 holds Note that A1 isimlied in Stes 1 and by the rincile of induction his comletes the roof

13 he Scientific World Journal 13 B he Proof of Lemma 5 We rove the conclusion by induction for the ositive integer k or k1wehavethat tr X j X j 1 S j 1 tr X j X j 1 [ 1 A ij M i tr X j X j 1 [ tr M i P j A ij M i A ij M i tr diag M 1 A il X l 1 B il A il X l 1 B il B ij A il X l 1 B il B ij P j] A il X l 1 B il B ij ] A ij X j X j 1B ij M diag M tr diag M 1 A 1l X l 1 B 1l A l X l 1 B l A l X l 1 B l A 1j X j X j 1B 1j A j X j X j 1B j A j X j X j 1B j A 1l X l 1 B 1l R 1 M M diag M 1 M A l X l 1 B l M A l X l 1 B l A 1j X j 1 B 1j A j X j 1 B j A j X j 1 B j B1 Assume that 9 holds for kmwhen km1by Algorithm 3wehavethat tr X j X j m1 S j m1 tr X j X j m1 [ 1 A ij M i P j A ij M i R m1 R m S j m] tr X j X j m1 [ R m1 R m A ij M i tr X j X j m1 S j m tr M i A il X l m1 B il A il X l m1 B il B ij A il X l m1 B il B ij P j A il X l m1 B il B ij ]

14 14 he Scientific World Journal R m1 R m R m1 S j m A ij X j X j m1b ij tr diag M 1 M diag M tr X j X j m S j m tr S j m S j m R m1 R m R m R m1 S j m A 1l X l m1 B 1l A l X l m1 B l A l X l m1 B l A 1j X j X j m1b 1j A j X j X j m1b j A j X j X j m1b j S j m tr diag M 1 M M diag M 1 M M A 1l X l m1 B 1l A l X l m1 B l A l X l m1 B l A 1j X j m1 B 1j A j X j m1 B j A j X j m1 B j R m1 R m1 R m1 B herefore 9 holdsfork m 1 hus9 holdsbythe rincial of induction his comletes the roof Conflict of Interests he authors declare that there is no conflict of interests regarding the ublication of this aer he authors of the aer do not have a direct financial relation that might lead to a conflict of interests for any of the authors Acknowledgments he authors are grateful to the anonymous referee and Profs R Camoamor-Stursberg I Ivanov and Y Shi for their constructive and helful comments his work was artially suorted by National Natural Science und Oen und of Geomathematics Key Laboratory of Sichuan Province scsxdz01001 Key Natural Science oundation of Sichuan Education Deartment 1ZA008 the young scientific research backbone teachers of CDU KYGG01309 basical and alicational roject of Sichuan Provincial Deartment of science and technology 013JY0061 References [1] Ding and Chen On iterative solutions of general couled matrix euations SIAM Journal on Control and Otimization vol 44 no [] Ding and Chen Gradient based iterative algorithms for solvingaclassofmatrixeuations IEEE ransactions on Automatic Controlvol50no [3] Ding P X Liu and J Ding Iterative solutions of the generalized Sylvester matrix euations by using the hierarchical identification rincile Alied Mathematics and Comutationvol197no [4] J Ding Y Liu and Ding Iterative solutions to matrix euations of the form A i B i i Comuters and Mathematics with Alications vol 59 no [5] L Xie J Ding and Ding Gradient based iterative solutions for general linear matrix euations Comuters and Mathematics with Alicationsvol58no [6] Ding and Chen Iterative least-suares solutions of couled Sylvester matrix euations Systems and Control Letters vol 54 no [7] AGWuGengGRDuanandWJWu Iterativesolutions to couled Sylvester-conjugate matrix euations Comuters and Mathematics with Alications vol60no [8] A G Wu B Li Y Zhang and G R Duan inite iterative solutions to couled Sylvester-conjugate matrix euations Alied Mathematical Modelling vol35no [9] A G Wu G eng G R Duan and W J Wu inite iterative solutions to a class of comlex matrix euations with conjugate

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