Riccati Equation Solution Method for the Computation of the Solutions of X + A T X 1 A = Q and X A T X 1 A = Q

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1 he Open Appled Inforatcs Journal, 009, 3, -33 Open Access Rccat Euaton Soluton Method for the Coputaton of the Solutons of X A X A Q and X A X A Q Mara Ada *, Ncholas Assas, Grgors zallas and Francesca Sanda Departent of Coputer Scence and Boedcal Inforatcs, Unversty of Central Greece, Laa 3500, Greece Departent of Electroncs, echnologcal Educatonal Insttute of Laa, Laa, Greece Abstract: A test s developed for checng the exstence of a fnte nuber of solutons of the atrx euatons X A X A Q and X A X A Q, when A s a nonsngular atrx An algebrac ethod for coputng all the solutons of these atrx euatons s proposed he ethod s based on the algebrac soluton of the correspondng dscrete te Rccat euatons he nuber of solutons s also derved he extree solutons are coputed as well Keywords: Matrx theory, atrx euatons, egenvalue-egenvector, Rccat euaton INRODUCION he central ssue of ths paper s the nvestgaton and the coputaton of all the solutons of the atrx euatons X A X A Q () and X A X A Q () where A s an n n atrx, A denotes the transpose of A and Q s an n n Hertan postve defnte atrx hese euatons arse n any applcatons n varous research areas ncludng control theory, ladder networs, dynac prograng, stochastc flterng and statstcs: see [, ] for references concernng euaton () and [3] for references concernng euaton () Concernng euaton (), we denote a soluton of X A X A Q as X It s well nown [4, heore 5] that, the exstence of the Hertan postve defnte soluton of the euaton () depends on the nuercal radus / / of atrx Q AQ In partcular, t s reured that / / rq ( AQ ), where r( A) ax{ x Ax : for every x n wth x x } denotes the nuercal radus of A It s also nown [4-6] that, f () has a Hertan postve defnte soluton X, then there exst nal and axal solutons X n and X ax, respectvely, such that 0 < Xn X Xax for any Hertan postve defnte soluton X Here, f X *Address correspondence to ths author at the Departent of Coputer Scence and Boedcal Inforatcs, Unversty of Central Greece, Laa 3500, Greece; E-al: ada@ucggr Matheatcs Subect Classfcaton: 5A4, 5A8, 93B5 and Y are Hertan atrces, then X Y eans that Y X s a nonnegatve defnte atrx and X < Y eans that Y X s a postve defnte atrx Concernng euaton (), we denote a soluton of X A X A Q as X It s well nown [3, 6] that there always exsts a unue Hertan postve defnte soluton, whch s the axal soluton X ax of (), and, f A s nonsngular, then there exsts a unue Hertan negatve defnte soluton, whch s the nal soluton X n of () he nal and axal solutons are referred as the extree solutons hese euatons have been studed recently by any authors [-]: the theoretcal propertes such as necessary and suffcent condtons for the exstence of a postve defnte soluton have been nvestgated and nuercal ethods for coputng the extree solutons of these euatons have been proposed; the avalable ethods are recursve algorths based anly on the fxed pont teraton and on applcatons of the Newton s algorth or the cyclc reducton ethod Furtherore, these euatons ay have Hertan postve defnte solutons or non- Hertan solutons hey also ay have defnte or ndefnte nuber of solutons For exaple as referred n [], when A Q 0 0, the euaton () has ndefnte nuber of solutons fored as X a a, where a s any real nuber a 0 Fnally, the exstence of a fnte nuber of Hertan postve defnte solutons of euaton () has been studed n [4], when A s nonsngular and Q I, where I s the dentty atrx In fact, the exstence of a fnte nuber of solutons of X A X A I depends on atrx A and s calculated, when A s a nonsngular coplex atrx / Bentha Open

2 Rccat Euaton Soluton Method he Open Appled Inforatcs Journal, 009, Volue 3 3 or A s a noral atrx ( AA A A ) Also, the fnte nuber of real syetrc postve defnte solutons of X A X A I s calculated, when A s real and nvertble In ths paper, we focus n the coputaton of all the solutons of the atrx euatons () and (), when A s nonsngular In Secton, an algebrac ethod for coputng all the solutons of these euatons s proposed he ethod s based on the algebrac soluton of the correspondng dscrete te Rccat euatons he nuber of solutons s also derved In Secton 3, a test for checng the exstence of a fnte nuber of solutons s developed he nuber of Hertan, syetrc, real and the extree solutons are coputed as well In Secton 4, sulaton results are gven to llustrate the effcency of the proposed ethod and fnally, concludng rears are gven n Secton 5 RICCAI EQUAION SOLUION MEHOD FOR HE COMPUAION OF ALL HE SOLUIONS All the Solutons of Euaton X A X A Q Here, A s an n n nonsngular atrx and the nuercal radus / / rq ( AQ ) orng as n [7] and [4], we are able to derve a Rccat euaton, whch s euvalent to the atrx euaton () In fact, the atrx euaton () can be wrtten as whereby arses, X Q A X A X QA [ QA X A] A QA A [ A QA X ] A A Q A A [ X ( A QA )] ( A A ) and thus the followng euvalent related Rccat euaton s derved ( ) ( ) P C F P G F, (3) wth F A A, C Q and G A QA (4) It becoes obvous that the atrx euaton () s euvalent to the related Rccat euaton (3) and that the two euatons have euvalent solutons hus the unue axal soluton of () concdes wth the unue Hertan postve defnte soluton of the related Rccat euaton: X P (5) ax It s clear that we are able to solve (), f we now the soluton of the related Rccat euaton he soluton of the related Rccat euaton can be derved usng the algebrac soluton proposed n [3] and [4] More specfcally, fro the Rccat euaton s paraeters the followng syplectc atrx s fored: A A A QA QA A A A QA QA Note that a syplectc atrx M JM J, where J O I I O (6) M s defned as Snce s a syplectc atrx, t can be wrtten as L ( ), (7) where all the egenvalues of are non-zero and can be placed n the dagonal atrx L O O and s the atrx, whch contans the correspondng egenvectors of atrx, (9) Rewrtng the eualty n (7) as L, we conclude the followng eualtes: A A ( A QA ) (0) ( ) QA A A QA A A () A A ( A QA ) () ( ) QA A A QA A A (3) Snce all the egenvalues of the syplectc atrx are non-zero, t s clear that 0 ( ) and 0 ( ), where ( A) denotes the spectru of A Moreover, f the bloc atrces,,, are nonsngular, then, after soe algebra, fro (0) and () we tae: ( ) Q A A ( ) A QA Fro () and (3) we tae: ( ) Q A A ( ) A QA A A A A (8) (4) (5) Usng (4) n (4) and (5), we are able to rewrte the last two euatons as ( ) C F ( ( ) ) G ( ) C F ( ( ) ) G (F ), (F ),

3 4 he Open Appled Inforatcs Journal, 009, Volue 3 Ada et al whereby t s obvous that both the uanttes ( ) and ( ) satsfy the Rccat euaton (3) and conseuently satsfy the atrx euaton () hus, the followng proposton s proved Proposton Let the bloc atrces,,, n (9) be nonsngular and arse fro every perutaton of coluns of n (9), whch consst of the egenvectors of n (6) he solutons of () are fored by ( ) X (6) X ( ) (7) At ths pont, we observe that the solutons calculated by (6) can be derved fro any arrangeent of the frst n egenvectors and that the other solutons calculated by (7) can be derved fro any arrangeent of the next n egenvectors Proposton he solutons n (6), (7) of euaton () do not depend on the perutaton of the frst n coluns of n (9), whch are egenvectors of n (6) Proof Let E rt O 0 0 O (8) be the n n dentty atrx, n whch the t colun s perutated by the r colun Let be the syplectc atrx n (6) and,,, n, n,, n be the egenvectors of n (7), where n n n and n n n Suppose that X, X consttute two solutons of () and due to (6), (7), they are wrtten X ( ) and X ( ) respectvely Snce E E and E E we tae X ( ) E E ( ) ( E )( E ) ( ) and ( ) ( ) E E X E E ( )( ) ( ) Conseuently, every perutaton of the frst n egenvectors consttutes the sae soluton for () and every perutaton of the followng n egenvectors consttutes the sae soluton for () Furtherore, we are gong to nvestgate every perutaton of the egenvalues of n a dagonal atrx In fact, as referred n [4], the syplectc atrx can be wrtten as ˆ ˆ ˆ L ( ), (9) where ˆL s the atrx, whch contans all the egenvalues of atrx as L n (8) and forulated as ˆL ˆ O O ( ˆ ), (0) wth ˆ the dagonal atrx, whch contans the egenvalues of atrx, that le outsde the unt crcle Note that the egenvalues of atrx occur n recprocal pars, due to the fact that s a syplectc atrx In (9), ˆ s the atrx, that contans the egenvectors of atrx, whch correspond to the egenvalues arranged n the dagonal atrx ˆL n (0): ˆ () Usng the atrx of egenvectors ˆ n (), Proposton and (6) of Proposton, we have: X P () ( ) Moreover, by (5) and (), the unue axal soluton of () s coputed as

4 Rccat Euaton Soluton Method he Open Appled Inforatcs Journal, 009, Volue 3 5 ˆ ˆ Xax ( ), that has been proved n a dfferent way n [7] By (7) the other soluton of () s coputed X ( ), whch s the nal soluton of (), as proved n [7]: X n ( ) Conseuently, the extree solutons X ax, X n X A X A Q are derved, dong the specfc arrangeent n the atrces L,, whch are fored as n (0)-() In the followng, by Proposton and, we lead to the coputaton of the nuber of solutons concernng euaton () heore 3 Let A be an n n nonsngular atrx, wth / / rq ( AQ ), and the bloc atrces,,, be nonsngular, whch arse fro every perutaton of coluns of n (9) he fnte nuber of solutons (ns) of () s eual to ( n)! ns (3) / / Proof Snce rq ( AQ ), the euaton () has a Hertan postve defnte soluton, (see [4, heore 5]) and by [4] the atrx n (6) for a specfc arrangeent of egenvalues (recall the egenvalues belong outsde or nsde the unt crcle) can be wrtten as n (9), t s obvous that, the euaton () has at least a Hertan postve defnte soluton, naely the axal one Moreover, the syplectc atrx n (7) has n egenvectors, whch can be used to copute all the solutons by (6)-(7) he fnte nuber of the possble perutatons of these egenvectors are (n)! By Proposton each soluton has ultplcty eual to n Hence, the nuber of solutons of () coputed usng the Rccat euaton soluton ethod s: ns n n (n)! n!(nn)! (n)! n!n! (n)! (n!) Rear Note that n heore 3, f we ot the nonsngularty of all the bloc atrces,,, whch arse fro every perutaton of the n frst wth the n followng coluns of n (9), then ns n (3) s an upper bound for the nuber of solutons of () Recall that the exstence of the solutons of () depends on / / rq ( AQ ) Conseuently, the exstence of the extree solutons and the nonsngularty of the correspondng atrces, are ensured by the assuptons of heore 3, therefore the solutons are of coputed by (6)-(7) he nonsngularty of all the bloc atrces,,, s needed for the exstence of all the other (not extree) solutons of (), whch are gven by forulas n (6)-(7) All the Solutons of Euaton X A X A Q Let A be a nonsngular atrx orng as n [7] and [3], we are able to derve a Rccat euaton, whch s euvalent to the atrx euaton () In fact, the atrx euaton () can be wrtten as whereby arses, X Q A X A X Q A [ Q A X A] A Q A A [ A QA X ] A A Q A A [ X A QA ] ( A A ) and thus the followng euvalent related Rccat euaton s derved ( ) ( ) P C F P G F (4) wth F A A, C Q and G A QA (5) It becoes obvous that the atrx euaton () s euvalent to the related Rccat euaton (4) and that the two euatons have euvalent solutons hus the unue axal soluton of () concdes wth the unue Hertan postve defnte soluton of the related Rccat euaton: X ax P (6) It s clear that we are able to solve (), f we now the soluton of the related Rccat euaton he soluton of the related Rccat euaton can be derved usng the algebrac soluton proposed n [3] and [4] More specfcally, fro the Rccat euaton s paraeters the followng syplectc atrx s fored: A A A QA (7) QA A A A QA QA Snce s a syplectc atrx, t can be wrtten as L ( ), (8) where all the egenvalues of are non-zero and can be placed n the dagonal atrx L O, (9) O and s the atrx, whch contans the correspondng egenvectors of atrx, (30)

5 6 he Open Appled Inforatcs Journal, 009, Volue 3 Ada et al Rewrtng the eualty n (8) as L, and followng the sae way of proof as n paragraph, we conclude : ( ) ( ) Q A A ( ) A QA Q A A ( ) A QA A A A A (3) (3) Usng (5) n (3) and (3), t s obvous that both the uanttes ( ) and ( ) satsfy the Rccat euaton (4) and conseuently satsfy the atrx euaton () hus, the followng proposton s proved n Proposton 4 Let the bloc atrces,,, (30) be nonsngular and arse fro every perutaton of coluns of n (30), whch consst of the egenvectors of n (7) he solutons of () are fored by ( ) X (33) ( ) X (34) At ths pont, we observe that the solutons calculated by (33) can be derved fro any arrangeent of the frst n egenvectors and that the other solutons calculated by (34) can be derved fro any arrangeent of the next n egenvectors hus, followng the sae ethod as n paragraph, the followng proposton s proved Proposton 5 he solutons n (33), (34) of euaton () do not depend on the perutaton of the frst n coluns of n (30), whch are egenvectors of n (7) Furtherore, n the proof of the forulas of the extree solutons n [7], we dd not observe that these are derved by the specfc perutaton of the egenvalues of the dagonal atrx In fact, as referred n [4], the syplectc atrx can be wrtten as ˆ ˆ ˆ L ( ), (35) where ˆL s the atrx, whch contans all the egenvalues of atrx as L n (9) and forulated ˆL ˆ O O ( ˆ ) (36) wth ˆ the dagonal atrx, whch contans the egenvalues of atrx, that le outsde the unt crcle In (35), ˆ s the atrx, that contans the egenvectors of atrx, whch correspond to the egenvalues arranged n the dagonal atrx ˆL n (36): (37) It s obvous that, usng the atrx of egenvectors ˆ n (37), Proposton 5, the forulas n (33)-(34) of Proposton 4, the unue axal and nal solutons of () are coputed as: X (38) ax ( ) X (39) n ( ) Conseuently, the extree solutons X ax, X n X A X A Q are derved, dong the specfc arrangeent n the atrces L,, whch are fored as n (36)-(37) By Proposton 4 and 5 and followng the sae way of the proof as n paragraph, we lead to the coputaton of the nuber of solutons concernng euaton () heore 6 Let A be an n n nonsngular atrx and the bloc atrces,,,, whch arse fro every perutaton of coluns of n (30), be nonsngular he fnte nuber of solutons (ns) of () s eual to ( n)! ns Rear Note that n heore 6, f we ot the nonsngularty of all the bloc atrces,,,, whch arse fro every perutaton of the n frst wth the n followng coluns of n (30), then ns n (3) s an upper bound for the nuber of solutons of () Recall that for an A nonsngular atrx, the euaton () has always extree Hertan solutons, whereby the nonsngularty of the correspondng atrces, s pled, therefore the solutons are coputed by (33) and (34) or by (38)-(39) he nonsngularty of all the bloc atrces,,,, s needed for the exstence of all the other (not extree) solutons of (), whch are gven by forulas n (33)-(34) 3 EXISENCE OF A FINIE NUMBER OF SOLUIONS 3 Exstence of a Fnte Nuber of Solutons for XA X A Q he exstence of a fnte nuber of solutons of euaton (), when A s nonsngular and Q I, depends on atrx A, as stated n [4] In fact, t depends on the egenvalues of the atrx: of

6 Rccat Euaton Soluton Method he Open Appled Inforatcs Journal, 009, Volue 3 7 H O A A A (40) More precsely: f d( V( ( H))) for all egenvalues ( H ),,,, n of H, wth ( H ), then, there exsts a fnte nuber of Hertan solutons of X A X A I Note that V( ( H)) denotes the egenspace correspondng to egenvalue ( H ) of the atrx H and d( V( ( H))) denotes the denson of V( ( H)) Moreover, the nuber of Hertan postve defnte solutons (hpdns) of X A X A I s gven by [4] hpdns ( n ), (4) where s the nuber of egenvalues ( H ) of H lyng outsde the unt crcle, wth algebrac ultplcty n,,,, Also, as referred n [4, Proposton 8], the nuber of real syetrc postve defnte solutons (rpdns) s eual to rpdns ( n ), (4) where p s the nuber of real egenvalues lyng outsde the unt crcle, s the nuber of coplex conugate pars of egenvalues lyng outsde the unt crcle, wth algebrac ultplcty n,,,, hs result, that s referred n the specal case, where Q I, can be extended for the general case, when Q I It s nown [4] that the atrx euaton () can be reduced to a correspondng atrx euaton, wth Q I Multplyng on / the rght and on the left both sdes of () by Q, (recall that Q s a Hertan postve defnte atrx, hence t s nonsngular), we have : / / / / Q X Q Q A X AQ I Settng n the last euaton / / Y Q XQ and we obtan / / R Q AQ (43) Y R Y R I, (44) where the atrx R s nonsngular, snce A s nonsngular By (43), t s obvous that the soluton X of () can be coputed as / / X Q Y Q, (45) where Y s the correspondng Hertan soluton of (44) Snce the exstence of a fnte nuber of Hertan postve defnte solutons of (44) depends on the atrx, whch s fored as n (40), we have H O R R R, and substtutng the atrx R by (43) we obtan: H O Q / A Q / Q / A Q / Q / A Q / (46) It s easy to verfy that the atrx H s syplectc Usng the above stateents we conclude the followng results, whch consttute a generalzaton of Corollary 66 and of Proposton 8 n [4] as well as of the relatonshps n (4)-(4) Corollary 7 Suppose that the atrx euaton X A X A Q has a Hertan postve defnte soluton If d( V( ( H ))) for every egenvalue ( H ),,,, n of H n (46), wth ( H ), then there exsts a fnte nuber of solutons of () Let ( H ), ( H ),, ( H ) be the dstnct egenvalues of H lyng outsde the unt crcle, wth algebrac ultplcty n,,,, he nuber of Hertan postve defnte solutons (hpdns) of () s eual to hpdns ( n ) (47) Corollary 8 Let A be an n n nonsngular real atrx, Q be real and H be the atrx n (46) Let ( H ), ( H ),, p ( H ) be the dstnct real egenvalues of H lyng outsde the unt crcle, be the nuber of coplex conugate pars of egenvalues lyng outsde the unt crcle, wth algebrac ultplcty n,,,, he nuber of real syetrc postve defnte solutons (rpdns) of () s eual to rpdns ( n ) (48) he exstence of a fnte nuber of solutons of the atrx euaton () can be expressed wth respect to the egenvalues of nstead of the egenvalues of H, as t s presented n the followng heore 9 Let A be an n n nonsngular atrx, wth / / rq ( AQ ), be the atrx n (6) and the bloc atrces,,, n (9) be nonsngular If d( V ( ( ))) for every egenvalue ( ),,,, n of, wth ( ), then, there exsts a

7 8 he Open Appled Inforatcs Journal, 009, Volue 3 Ada et al fnte nuber of Hertan postve defnte solutons (hpdns) of () he nuber of Hertan postve defnte solutons of () s eual to hpdns ( n ) (49) he nuber of non-hertan solutons (nhns) of () s nhns ns ( n ) (50) In partcular, f A and Q are real atrces, then aong the hpdns there exst real syetrc solutons (rpdns) and the nuber of rpdns of () s rpdns ( n ) (5) In above, s the nuber of the dstnct egenvalues of, that le outsde the unt crcle, wth algebrac ultplcty n,,,,, p s the nuber of real dstnct egenvalues of lyng outsde the unt crcle, s the nuber of coplex conugate pars of egenvalues lyng outsde the unt crcle, wth algebrac ultplcty n,,,,, and ns s gven by (3) Proof Due to the followng relaton Q/ O O Q / ((H ) ) Q / O O Q /, (5) t s clear that the atrces and ( H ) are slar, thus the relaton between the egenvalues of these two atrces s: H ( ) ( ),,,, n (53) Moreover, by (53), the relaton of the absolute value s: ( H ) ( ) By the last eualty t s obvous that, ( H ) ( ) (54) and ( H ) > ( ) > (55) Note that the egenspaces, correspondng to egenvalues ( H ) and ( H ) of the atrces H and ( H ), concde, hence, d( V( ( H ))) d( V( ( H ))) Also, f we cobne the above eualty wth the slarty of atrces and ( H ), we conclude that d( V( ( ))) d( V( ( H ))) (56) he assupton and (56) yelds d( V( ( ))) d( V( ( H ))) he last stateent and (54) lead to the concluson that, the frst part of Corollary 7 can be appled, e, there exsts a fnte of Hertan solutons of () Furtherore, cobnng the assupton wth (55), t s obvous that the second part of Corollary 7 s appled, hence, hpdns ( n ), where n s the algebrac ultplcty of the egenvalue ( ) of, for every,,,, wth ( ) > Also, by (53) we have that any real (coplex) egenvalue ( H ) of the atrx H s real (coplex) egenvalue of the atrx, respectvely, thus the hypotheses and (55) lead to applcaton of Corollary 8, whereby arses the nuber of real syetrc solutons n (5) Furtherore, due to the assuptons for the bloc atrces,,,, the nuber of non-hertan solutons of () n (50) yelds cobnng heore 3 wth the euaton (49) Rear 3 Note that n heore 9, p,, are connected by the relatonshp 3 Exstence of a Fnte Nuber of Solutons for X A X A Q he euaton () can be fored as (), f we substtute the nonsngular atrx A wth A, and then we have X A X A Q (57) Accordng to heore 9, the exstence of a fnte nuber of solutons of euaton (57) depends on the atrx, whch s fored as (6) and s A A QA A A QA A A QA QA (58) heore 0 Let A be an n n nonsngular atrx be the atrx n (7) and the bloc atrces,,, n (30) be nonsngular If d( V ( ( ))) for every egenvalue ( ),,,, n of, wth ( ), then, there exsts a fnte nuber of Hertan solutons (hns) of () he nuber of Hertan solutons of () s eual to hns ( n ), (59) and the nuber of non-hertan solutons (nhns) of () s nhns ns ( n ) (60) In partcular, f A and Q are real atrces, then aong the hns there exst real syetrc solutons (rns) and the nuber of rns of () s

8 Rccat Euaton Soluton Method he Open Appled Inforatcs Journal, 009, Volue 3 9 rns ( n ) (6) In above, s the nuber of the dstnct egenvalues of, that le outsde the unt crcle, wth algebrac ultplcty n,,,,, p s the nuber of real dstnct egenvalues of lyng outsde the unt crcle, s the nuber of coplex conugate pars of egenvalues lyng outsde the unt crcle, wth algebrac ultplcty n,,,,, and ns s gven by (3) Proof It s obvous that, when we substtute n (58) A wth A, arses A A A QA, (6) QA A A A QA QA e, the atrx s dentfed wth the atrx n (7) Hence, t becoes obvous that the soluton of () can be derved through the soluton of euaton of type () n (57) and cobnng the conclusons of heore 9 In fact, f d(v ( ( ))) d(v ( ( ))), for all egenvalues ( ) of the atrx wth ( ) ( ),,,, n, then there exsts a fnte nuber of Hertan solutons of X A X A Q, due to the exstence of fnte nuber of Hertan postve defnte solutons of X A X A Q Clearly, snce ( ) ( ),,,, n, the egenvalues ( ), ( ),, ( ) le outsde the unt crcle, when ( ), ( ),, ( ) le outsde the unt crcle Also, any real (coplex) egenvalue ( ) of s real (coplex) egenvalue of ; therefore, applyng the conclusons of heore 9 for the atrx, the eualtes n (59) and (6) are derved, edately Furtherore, due to the assuptons for the bloc atrces,,,, the nuber of non-hertan solutons of () n (60) yelds cobnng heore 6 wth the euaton (59) Rear 4 Note that n heore 0, p,, are connected by the relatonshp 4 SIMULAION RESULS Sulaton results are gven to llustrate the effcency of the proposed ethod he proposed ethods copute accurate solutons as verfed through the followng sulaton exaples, usng Matlab 65 Exaple hs exaple concerns the scalar euaton X A X A Q, wth A 05, Q Snce A s a nonsngular atrx, Rccat euaton soluton ethod can be appled he atrx n (6) has ( ) {0078, 398}, and the correspondng egenvectors are the coluns of the atrx Conseuently, has egenvalue outsde the unt crcle, whch s a real egenvalue p, wth ultplcty n and no par of coplex egenvalues ( 0 ) Snce all the hypotheses of heore 9 are verfed, by (49) and (5) the nuber of Hertan postve defnte solutons s eual to the nuber of real syetrc postve defnte solutons: hpdns ( n ) rpdns ( n ) Furtherore, n and by (3) the ethod coputes ( n)! ns solutons of the atrx euaton X A X A Q and accordng to heore 9 the nuber of non-hertan solutons s gven by (50), e, the non- Hertan solutons are nhns ns ( n ) 0 hus, there exst only Hertan-real syetrc solutons, whch are the extree solutons: X X ax (axal soluton) X X n (nal soluton) It s also verfed that: O < Xn < Xax Exaple hs exaple concerns the euaton X A X A Q and s taen fro [] Consder the euaton (), wth A 3 4, Q Snce A s a real and nonsngular atrx wth / / rq ( AQ ) 04957, Rccat euaton soluton ethod can be appled he atrx n (6) has, ( ) { ( ) 053± 08506, ( ) 0457 ± 07 }, 3,4 wth, ( ) 3, 3,4 ( ) 045, and the correspondng egenvectors of are the coluns of the atrx

9 30 he Open Appled Inforatcs Journal, 009, Volue 3 Ada et al he atrx has egenvalues outsde the unt crcle, p 0 real egenvalue, par of coplex egenvalues outsde the unt crcle, the algebrac ultplcty of all the egenvalues s n n Moreover, the bloc atrx,,,, that arses fro every perutaton of coluns of, s nonsngular hus, the hypotheses of heore 9 are verfed and by (49) we copute hpdns ( n ) 4, and by (5) rpdns ( n ) Furtherore, by (3) of heore 3, for n, the ( n)! ethod coputes ns 6 solutons of the atrx euaton X A X A Q Accordng to heore 9 the nuber of non-hertan solutons s gven by (50) and s eual to nhns ns ( n ) 6 4 All the solutons are: X X ax (axal soluton) X X X X X 6 X n (nal soluton) It s also verfed that: O < Xn < Xax Exaple 3 hs exaple concerns the euaton X A X A Q and s taen fro [] Consder the euaton (), wth A , Q Snce A s a real and nonsngular atrx wth / / rq ( AQ ) 04998, Rccat euaton soluton ethod can be appled he atrx n (6) has ( ) {, ( ) 785 ± 00583, 3 ( ) 0673, ( ) 0937, ( ) 007± }, 4 5,6 wth, ( ) 055, 3 ( ) 0673, 4 ( ) 0937, and 5,6 ( ) he correspondng egenvectors of are the coluns of the atrx 3 he atrx has 3 egenvalues outsde the unt crcle, fro that, p real egenvalue, par of coplex egenvalues outsde the unt crcle, the algebrac ultplcty of all the egenvalues s eual to Moreover, the 3 3 bloc atrx,,,, that arses fro every perutaton of coluns of 3, s nonsngular hus, the hypotheses of heore 9 are verfed and we copute hpdns ( n ) 8 and rpdns ( n ) 4 by (49) and (5), respectvely he real syetrc postve defnte solutons are X, (axal soluton), X, X 9, X 0, (nal soluton), n partcular: X X ax

10 Rccat Euaton Soluton Method he Open Appled Inforatcs Journal, 009, Volue X X X 0 X n It s verfed that: O < Xn X Xax for every Hertan postve defnte soluton X, snce n, 9 ax O < X X X X Furtherore, by (3) of heore 3, for n 3, the ( n)! ethod coputes ns 0 solutons of the atrx euaton X A X A Q Accordng to heore 9 the nuber of non-hertan solutons s gven by (50) and s eual to nhns ns ( n ) 0 8 Exaple 4 hs exaple concerns the scalar euaton X A X A Q, wth A, Q Snce A s a nonsngular atrx, Rccat euaton soluton ethod can be appled he atrx n (7) has ( ) {06096, 6404}, and the correspondng egenvectors are the coluns of the atrx Conseuently, has egenvalue outsde the unt crcle, whch s a real egenvalue p, wth ultplcty n and 0 Snce all the hypotheses of heore 0 are verfed, by (59) and (6) the nuber of Hertan solutons and especally of real solutons s eual to hns ( n ), rns ( n ), respectvely Furtherore, n and by heore 6, the ( n)! ethod coputes ns solutons of the atrx euaton X A X A Q Accordng to heore 0 the nuber of non-hertan solutons s gven by (60) and s eual to nhns ns ( n ) 0 hus, there exst only Hertan-real syetrc solutons, whch are the extree solutons: X X ax 566 (axal soluton) X X n 566 (nal soluton) Exaple 5 hs exaple concerns the euaton X A X A Q and s taen fro [6] and [] Consder the euaton (), wth A , Q 3 4 Snce A s a real and nonsngular atrx, Rccat euaton soluton ethod can be appled he atrx n (7) has ( ) {, ( ) 0404 ± 098, ( ) 0976 ± 0766 }, 3,4 wth, ( ) 059, 3,4 ( ) 0944, and the correspondng egenvectors of are the coluns of the atrx 5 he atrx has egenvalues outsde the unt crcle, wth n n, where p 0 and, and the bloc atrces,,,, that arse fro every perutaton of coluns of 5, are nonsngular hus, the hypotheses of heore 0 are verfed and we copute hns ( n ) 4, rns ( n ), by (59) and (6), respectvely Furtherore, n and by ( n)! heore 6, the ethod coputes ns 6 solutons of the atrx euaton X A X A Q Accordng to

11 3 he Open Appled Inforatcs Journal, 009, Volue 3 Ada et al heore 0 the nuber of non-hertan solutons s gven by (60) and s eual to nhns ns ( n ) 6 4 All the solutons are: X X ax (axal soluton) X X X X X 6 X n (nal soluton) Exaple 6 hs exaple concerns the euaton X A X A Q, wth A , Q hs exaple has the sae atrces AQ, as Exaple 3, thus A s nonsngular, hence the Rccat euaton soluton ethod can be appled he atrx n (7) has, 3 4 5,6 ( ) { ( ) 74± 4359, ( ) 6644, ( ) 06, ( ) ± }, wth, ( ) 44644, 3 ( ) 6644, 4 ( ) 06 and 5,6 ( ) 0069 he correspondng egenvectors of are the coluns of the atrx 6 he atrx has 3 egenvalues outsde the unt crcle, wth n n n 3, fro that, there exsts p real egenvalue and par of coplex egenvalues outsde the unt crcle Moreover, the 3 3 bloc atrx,,,, that arses fro every perutaton of coluns of 6, s nonsngular hus, the hypotheses of heore 0 are verfed and we copute hns ( n ) 8 and rns ( n ) 4 by (59) and (6), respectvely he real syetrc solutons are X, (axal soluton), X, X 9, X 0, (nal soluton), n partcular: X X ax X X 9 X 0 X n Furtherore, n 3 and by heore 6, the ethod ( n)! coputes ns 0solutons of the atrx euaton X A X A Q Accordng to heore 0 the nuber of non-hertan solutons s gven by (60) and s eual to nhns ns ( n ) 0 8 CONCLUSIONS A ethod s developed for checng the exstence of a fnte nuber of solutons of the atrx euatons

12 Rccat Euaton Soluton Method he Open Appled Inforatcs Journal, 009, Volue 3 33 X A X A Q and X A X A Q, when A s nonsngular An algebrac ethod for coputng of all the solutons of these atrx euatons s proposed, naely the Rccat euaton soluton ethod, whch s based on the algebrac soluton of the correspondng dscrete te Rccat euatons he ethod provdes sple forulas for coputng the accurate solutons of these atrx euatons, as verfed through sulaton experents he ethod coputes the extree solutons as well he nuber of solutons s also derved he exstence and the coputaton only of the extree solutons (Hertan) of the atrx euatons () and () have been studed n the bblography (see [3, 4, 7, 0]) In the present paper, the necessary condtons for the exstence of all the solutons (Hertan ones as well as non-hertan ones) of these euatons are presented, the nuber of solutons s coputed and forulas for coputng all these solutons are gven REFERENCES [] N Anderson, Jr D Morley, and GE rapp, Postve solutons to X A BX B*, Lnear Algebra Appl, vol 34, pp 53-6, 990 [] X Zhan, Coputng the extreal postve defnte solutons of a atrx euaton, SIAM J Sc Coput, vol 7, no 5, pp 67-74, Sep 996 [3] A Ferrante, and BC Levy, Hertan solutons of the euaton X QNX N*, Lnear Algebra Appl, vol 47, pp , 996 [4] JC Engwerda, ACM Ran, and AL Reboer, Necessary and suffcent condtons for the exstence of a postve defnte soluton of the atrx euaton X A*X A Q, Lnear Algebra Appl, vol 86, pp 55-75, 993 [5] I Gohberg, P Lancaster, and L Rodan, Matrces and Indefnte Scalar Product, Operator heory: Advances and Applcatons, O8 Brhäuser Verlag: Basel, 983 [6] B Men, Effcent coputaton of the extree solutons of X A* X A Q and X A*X A Q, Math Coput, vol 7, no 39, pp 89-04, Nov 00 [7] M Ada, N Assas, and F Sanda, Algebrac solutons of the atrx euatons X A X A Q and X A X A Q, Int J Algebra, vol, no, pp 50-58, 008 [Onlne] Avalable fro: http//www-harco/a [Accessed 0 Feb 008] [8] N Assas, F Sanda, and M Ada, Recursve solutons of the atrx euatons X A X A Q and X A X A Q, Appl Math Sc, vol, no 38, pp , 008 [Onlne] Avalable fro: [Accessed 0 Mar 008] [9] P Benner, and H Fassbender, On the soluton of the ratonal atrx euaton X Q LX L, EURASIP J Adv Sgnal Process, vol 007, pp -0, 007 Artcle ID 850, [Onlne] Avalable fro: 055/007/850 [Accessed Feb, 007] [0] JC Engwerda, On the exstence of a postve defnte soluton of the atrx euaton X A X A I, Lnear Algebra Appl, vol 94, pp 9-08, 993 [] CH Guo, and P Lancaster, Iteratve soluton of two atrx euatons, Math Coput, vol 68, no 8, pp , Aprl 999 [] X Zhan, and J Xe, On the atrx euaton X A X A I, Lnear Algebra Appl, vol 47, pp , 996 [3] BDO Anderson, and JB Moore, Optal Flterng, Englewood Clffs, New Jersey, USA: Prentce Hall nc, 979 [4] DR Vaughan, A nonrecursve algebrac soluton for the dscrete Rccat euaton, IEEE rans Autoat Cont, pp , 970 Receved: March 30, 009 Revsed: Aprl 9, 009 Accepted: May 07, 009 Ada et al; Lcensee Bentha Open hs s an open access artcle lcensed under the ters of the Creatve Coons Attrbuton Non-Coercal Lcense ( whch perts unrestrcted, non-coercal use, dstrbuton and reproducton n any edu, provded the wor s properly cted