Partial Ordering of Range Symmetric Matrices and M-Projectors with Respect to Minkowski Adjoint in Minkowski Space

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1 Advaces i Liea Algeba & Matix Theoy, 26, 6, IN Olie: IN Pit: X Patial Odeig of age ymmetic Matices ad M-Pojectos with espect to Mikowski Adjoit i Mikowski pace D Kishaswamy, Mohd aleem Loe Depatmet of Mathematics, Aamalai ivesity, Chidambaam, Idia How to cite this pape: Kishaswamy, D ad Loe, M (26) Patial Odeig of age ymmetic Matices ad M-Pojectos with espect to Mikowski Adjoit i Mikowski pace Advaces i Liea Algeba & Matix Theoy, 6, eceived: Octobe 6, 26 Accepted: Decembe 3, 26 Published: Decembe 6, 26 Copyight 26 by authos ad cietific eseach Publishig Ic This wok is licesed ude the Ceative Commos Attibutio Iteatioal Licese (CC BY 4) Ope Access Abstact I this pape, we obtai some ew chaacteizatios of the age symmetic matices i the Mikowski pace by usig the Block epesetatio of the matices These chaacteizatios ae used to establish some esults o the patial odeig of the age symmetic matices with espect to the Mikowski adjoit Futhe, we establish some esults egadig the patial odeig of m-pojectos with espect to the Mikowski adjoit ad maipulate them to chaacteize some sets of age symmetic elemets i the Mikowski pace All the esults obtaied i this pape ae a extesio to the Mikowski space of those give by A Headez, et al i [The sta patial ode ad the eigepojectio at o EP matices, Applied Mathematics ad Computatio, 28: , 22] Keywods Patial Ode, Mikowski Adjoit, Mikowski Ivese, age ymmetic, M-Pojectos Itoductio ad Pelimiaies Let us deote by M ( m, ) ( ) the set of m M ( ) fo M ( ) ( ) The symbols A *, A, matices ad whe m we wite, A, A, ( A ) ad N ( A ) deote the cojugate taspose, Mikowski adjoit, Mikowski ivese, Mooe-Peose ivese, age space ad ull space of a matix A espectively I deote the idemp tity matix of ode Futhe we deote by the set of all m-pojectios ie mp 2 { P: P P P } Also we use the covectio accodig to which Pw WW ad P w Ik WW Whee I is the idetity matix of suitable ode ad s will k deote the ak of the matices ad DOI: 4236/alamt26643 Decembe 6, 26

2 D Kishaswamy, M Loe Idefiite ie poduct is a scala poduct defied by * [ ] u, v u, Mv u Mv, () whee, deotes the covetioal Hilbet pace ie poduct ad M is a Hemitia matix This Hemitia matix M is efeed to as metic matix Mikowski pace is a idefiite ie poduct space i which the metic matix 2 is deoted by G ad is defied as G satisfyig G I ad G * G I G is called the Mikowski metic matix I case u ( u, u,, u ), the G is called the Mikowski metic teso ad is defied as Gu ( u, u,, u ) Fo detailed study of idefiite liea algeba efe to [] The mikowski ivese of a matix M ( m, ) ( ), itoduced by Meeakshi i [2], is the uique solutio to the followig fou matix equatios: [MI-]: X [MI-2]: XX X [MI-3]: ( X ) X [MI-4]: ( X) X Howeve ulike the Mooe-Peose ivese of a matix, the Mikowski ivese of a matix does ot exist always I [2], Meeakshi showed that the Mikowski ivese of a matix M ( m, ) ( ) exists if ad oly if k ( ) k ( ) k ( ), whee * GG 2 is called the Mikowski adjoit of the matix ad G ad G 2 ae the Mikowski metic matices of suitable ode m ad A matix M ( ) is said to be m-symmetic if ad is said to be G-uitay if ad oly if I I [3], Meeakshi itoduced the cocept of age symmetic matices i Mikowski pace ad developed the Mikowski ivese of the age symmetic matices ad some equivalet coditios fo a matix to be age symmetic A matix M ( ) is said to be age symmetic if ad oly if N( ) N( ) I [4], the authos poduced the ecessay ad sufficiet coditios fo the poduct of age symmetic matices to be age symmetic ad futhe showed that ay block matix i Mikowski space ca be expessed as the poduct of age symmetic matices I [5] the authos studied the age symmetic matices i elatio with thei Mikowski ivese ad m-pojectos ummaizig the equivalet coditios fo the defiitio of a age symmetic matix fom [3] [5] [6] the followig equivalet coditios will be used i the fothcomig esults: [-]: is age symmetic N N [-2]: ( ) ( ) [-3]: [-4]: ( ) ( ) [-5]: thei exist a G-uitay matix such that ( D ) Patial odes o matices has emaied the topic of iteest fo may authos i the aea of matix theoy ad geealized ivese Almost all authos who have woked o patial odeig of matices have fomulated the defiitio ivolvig diffeet kids of 33

3 D Kishaswamy, M Loe geealized iveses ad i paticula the Mooe-Peose Ivese esults ivolvig patial odes o matices i elatio with thei geealized ivese ae scatteed i the liteatue of the matix theoy ad geealized iveses fo istace see [7]-[9] Patial odeig o matices has a wide age of applicatios i diffeet fields which iclude electical etwoks, statistics, geealized iveses etc see [2] [2] [22] [23] Diffeet kids of patial odes o matices have bee studied which iclude ta patial odeig * itoduced by Dazi [24], mius patial ode itoduced by Hatwig [25], # hap patial ode itoduced by Mita [9], followed by left sta odeig * ad ight sta odeig * I [26], Puithavalli itoduced the patial odeig o matices i Mikowski space wt the Mikowski adjoit he studied the patial odeig, left patial odeig ad ight patial odeig wt the Mikowski adjoit o age symmetic matices he also established some equivalet coditios fo the evese ode law to hold i elatio to the patial odeig wt Mikowski adjoit Fom ([26], page 79), we have fo ay two matices, M ( m, ) ( ), is said to be below ude the patial ode wt Mikowski adjoit, deoted by, if oe of the followig equivalet coditio is satisfied: [PO-]: ad [PO-2]: ad [PO-3]: ( ) ( ) ad ( ) ( ) I ay of the above cases we say is pedecesso of o is successo of k k We will use the otatio M ( ) M to deote the set of all the matices of idex k I this pape we obtai some chaacteizatios of age symmetic matices ad utilize them to study the patial odeig of age symmetic matices wt the Mikowski adjoit i Mikowski space ad hece diffeet chaacteizatios of patial odes o age symmetic matices ae obtaied Fially we study the patial odeig o m-pojectos wt the Mikowski adjoit All the esults obtaied i this pape ae a extesio of those give i [27] to the Mikowski space 2 Popeties of age ymmetic Matices I this sectio we develop some popeties of age ymmetic matices by utilizig the epesetatio obtaied i coollay 26 i [5] Let, M ( ) be o-zeo age symmetic matices of ak ad s espectively The ad, accodig to the above metioed esult, ca be witte as D (2) ad D (3) whee ad ae G-uitay ad Theoem Let M ( ), D ad D ae ivetible matices of ode be such that is age symmetic The the fol- 34

4 D Kishaswamy, M Loe lowig statemets ae equivalet: 2 If is give by (2), the thee exists J M ( ) ad M M ( ) such that with DJ JD M Poof We coside the decompositio of the matix, accodig to the size of blocks of, as: L M Fom the statemet (i) of the theoem, we get D D L M L M This gives DJ JD, L ad K ad hece the esult follows If both the matices ad ae age symmetic, the we have the followig esult fo the commutativity Theoem 2 Let, M ( ) be age symmetic matices If L M The the followig statemets ae equivalet: J D J D D J D J 2 ( ) ( ) 3 ( ) ( ) D JD J JD J D Poof (i) (ii) Coside the epesetatios of ad give by (2) ad (3) es- pectively With give, we have L M D D DJD (4) L M Also Theefoe J LG ( ) GK G M G D J LG D GK G M G DJ D Fom Equatios (4) ad (5) we have DJD DJ D (6) (5) Pe multiplyig ad post multiplyig (6) by stitutig the matix epesetatio of ad ad we get espectively ad sub- 35

5 D Kishaswamy, M Loe J DJD DJ DJ GK DJD DJ DK Fom this equality, o usig the fact that ( ) ( ) D ad D ae osigula, we have JDK ad hece the equivalece J DJ D D J DJ, K DJ, ad follows (i) (iii) Fom L M, usig the fact that is G-uitay, we have J LG ad hece L M ubstitutig the e- GK G M G pesetatios of ad i the block epesetatio of give by (3) we have D J LG L M GK G M G JD J JD L G LDJ LDL G Futhemoe, doig some algeba we have, D JD J D JD L G JD J D ad LDJ D Theefoe the equality, o usig the fact that D, osigula, gives ( ) ( ) D JD J JD J D, LD J ad JD L Hece the equivalece follows D ad G ae Theoem 3 Let M ( ) be such that exists The the followig statemets ae equivalet: is age symmetic 2 ( ) ( ) 3 N( ) N( ) Poof (i) (ii) ice ad ae m-symmetic idempotets, i fact m- pojectos, o usig [-3], we have is age symmetic if ad oly if Also fom [MI-] ad [MI-2] we have ( ) ( ) ad Hece the equivalece follows N ( ) ( ) Theefoe ( ) ( ) (i) (iii) imilaly ( I ) ad ( I ) ( ) ( I ) ad N( ) ( ) lows Theoem 4 Let ( ) ae equivalet: M is age symmetic ae idempotets such that I Agai usig [-3], the esult fol- be a o zeo matix The the followig statemets 2 Thee exists a ivetible matix M ( ) ad L ( ) ( ) M M, such that 36

6 D Kishaswamy, M Loe E ( D ) with E D L M M 3 Thee exists a ivetible matix M ( ) ad L ( ) ( ) M, such that E ( D ) 2 with E L M Poof (i) (ii) sig [-4], thee exists a ivetible matix E M ( ) such that E We patitio E accodig to the blocks of such that Now vetible ad E L M E ( D ) D, gives E G is G-uitay L M, usig the fact that D is i-, the equivalece follows o the same lies as above Theoem 5 Let M ( ) be a ozeo matix The the followig statemets ae equivalet: is age symmetic 2 Thee exists a ivetible matix M M ( ) ad K M (, ) ( ) such that (i) (iii) Fom statemet (ii) of the Theoem 3 ad [-4], we have ( ) ( ) F D D M M F ( ) with 3 Thee exists a ivetible matix M ( ) ad K ( ) ( ) M, such that F ( D ) 2 with F M Poof The poof follows o the same lies as i the above theoem, usig the fact that two matices ad ae ow equivalet if ad oly if N( ) N( ) ad utilizig the statemet (iii) of Theoem 3 ad [-2] 3 Patial Odeig of age ymmetic Matices wt Mikowski Adjoit I this sectio some chaacteizatios of pedecessos of age symmetic matices ude the patial odeig wt Mikowski adjoit sig the equivaleces of the defiitio of Patial odeig wt Mikowski adjoit that is [PO-] ad, [PO-2], it ca be easily veified that,, ad ae m-symmetic Theoem 6 Let, M ( ) such that is a ozeo age symmetic matix The the followig statemets ae equivalet: 2 Thee exists J M ( ) such that 37

7 D Kishaswamy, M Loe with J D (7) Poof (i) (ii) We coside the followig block epesetatio of accodig to the block size of as: ad The L M J J LG L J K LG M GK J + G( ) M G ( ) L GK K + G( ) M G( ) M D LD Theefoe the equality gives J J L GL JD ad G K K + G( ) M G( ) M K ad M Also computig ad ad usig the equality, we get JJ KG K D J ad L Thus J J JD ad JJ D J ie, Howeve if is age symmetic ad metic eg coside the matices Example J D i i ad, whee i i emak If both the matices M ( ),, the eed ot be age sym- ae age symmetic ad, the usig the statemets [PO-], [PO-2] ad [-3], it ca be easily obseved that sig the epesetatios (3) ad (7) of ad espectively ad Theoem 683 fom [26], we have aothe equivalet coditio fo the patial odeig of age symmetic matices wt mikowski adjoit give by ad Futhemoe, is age symmetic, we have The ext esult gives some equivalet coditios fo a matix to be age symmetic whe is age symmetic ad is the successo of ad Theoem 7 Let M ( ), such that is a ozeo age symmetic matix, whee is give by (3) ad is give by (7) The the followig statemets ae equivalet: is age symmetic 2 JD DJ J D DJ 3 4 ( ) ( ) 5 ( J D ) ( J J J) D J J D D J J 6 J is age symmetic Poof (i) (ii) Fom emak, we have Now usig the facts that 38

8 D Kishaswamy, M Loe D D ; D beig ivetible ad is G-uitay ad substitutig the epesetatios of ad fom (3) ad (7) espectively i the above equality ad doig some simple algeba leads to JD DJ (ii) (iii) Fo, Agai usig emak ad substitutig the espective epesetatios of ad, the equivalece follows (ii) (iv) sig [PO-] ad substitutig the epesetatios of, ad, the equivalece follows afte some computatio O the same lies the equivaleces (ii) (v) ad (iii) (vi) follow by usig the emak ad statemets [PO-] ad [PO-2] The ext esult simila to Theoem 6 holds if we coside to be age symmetic ad decompose i tems of epesetatio fo Theoem 8 Let, M ( ) such that is a ozeo age symmetic matix The the followig statemets ae equivalet: 2 Thee exists M M ( ) such that D (8) M Poof The poof follows o the same lie as i Theoem 6 We agai ote that if ad is age symmetic, the eed ot be age symmetic Coside Example I the followig esult we establish some equivalet coditios fo whe is age symmetic ad Theoem 9 Let, M ( ) be give by (2) ad (8) espectively such that is a ozeo age symmetic matix ad The the followig statemets ae equivalet: is age symmetic 2 M is age symmetic 3 ( ) ( ) 4 ( ) ( ) D Poof (i) (ii) Fo, sice D is osigula ad is G- M D uitay, diect veificatio shows that Theefoe M I I ad MM beig age sym- M M metic, by [-3] we have This gives MM M M ad the equivalece follows (i) (iii) ice ad ad ae age symmetic, usig the obsevatio metioed i emak ie,, we have, the equivalece follows (iii) (i) ice ad is age symmetic, agai by the same fact that 39

9 D Kishaswamy, M Loe ad commute, usig (iii) ie, ( ) ( ) symmetic (i) (iv) Fom emak, we have, we get is age This gives Now usig the fact that is age symmetic the equivalece follows I the above esults we have used the commutativity of ad ad ad Howeve if we assume the coditios give i the above theoem with a additioal assumptio that ae also equivalet Theoem Let M ( ), the the coditios obtaied by itechagig ad, matix The the followig statemets ae equivalet: be age symmetic such that is a o zeo 2 Thee exists a G-uitay matix M ( ), D M ( ) ad M M s( ) D D such that ad M Poof (i) (ii) Coside the decompositio of give by (3) ie, D ice is age symmetic, theefoe by Theoem 6, thee exists J M ( ) such that with J D sig Theoem 7, we have J is age symmetic We coside the followig block epesetatio of J as D J J J J, whee J is G-uitay ad D ( ) J D, by Theoem 8, we ca fid M M s( ) M such that Thus M is osigula whe M Takig D J D, we have D DJ M, whee is G-uitay I (ii) (iiii) Follows at oce by diect veificatio 4 Patial Odeig of M-Pojectos is ivetible ice D D M J J J I this sectio we obtai some esults o patial odeig of m-pojectos wt Mikowski adjoit The followig esult fom [5], with two moe obvious coditios, will be used extesively i the fothcomig esults Lemma Let be age symmetic, the + P I P ( ) ( ) 2 P is idempotet 3 P P P P 4 P if ad oly if is osigula 5 P( D ) P the P P( I ) P 4

10 D Kishaswamy, M Loe 6 ak ( P ) ak ( ) 7 ad if, the I P is ivetible the P 8 P has idex atmost oe Lemma 2 Let, M M The If, the k ( ) k ( ) 2 P 3 P 4 If is ozeo sigula matix the ad P ae icompaable ude the patial odeig wt Mikowski adjoit 5 P P k k 6 + P Poof (i) ice ( ) ( ) ( ) ( ) ( ) This gives k ( ) k ( ) (ii) P, the P, ad P I ad hece P (iii) Fom statemet (ii) of Lemma ad the fact that ( P ) P, if, the P P ad hece by poit (iv) of Lemma is ivetible Agai by the same agumet ie, poit (iv) of Lemma covese holds (iv) It is obvious fom (ii) ad (iii) (v) Follows at oce by usig poit (i) of the Lemma 2 ad poit (vi) of Lemma (vi) The statemet follows at oce o usig the fact that P Lemma 3 Let M M The If is age symmetic, the P is m-symmetic ad hece age symmetic 2 If P is age symmetic, the so is Poof (i) The statemet follows at oce o usig the [-3], [MI-3] ad [MI-4] (ii) If, the the esult is tivial Let such that k ( P ), the by D poit (v) of Lemma we have P Also usig poit (ii) of Lemma we get D 2 D ie, D I Thus we have Coside the block e- I pesetatio of, whee the patitio is doe accodig to the blocks of P such that sig [MI-] ad [MI-2], we get J, K ad L Theefoe L M This shows that M is osigula ad the esult follows MM emak 2 ice P is a m-pojecto [5], we have ( P ) P If we wite P P ie, we take P as a fuctio of, the ( ) ( ) ( ) Thus P ( ) ( ) ( ) P2 P P I P P 2 Howeve P D 2 ( ) i geeal Coside the decompositio, we have 4

11 D Kishaswamy, M Loe P 2 ( ) D I ie, I, which is the fudametal epese- P if ad oly if is a m- tatio of a m-pojecto Hece we coclude that ( ) pojecto We geealize the fuctio P( ) Thus we have the followig equatios by defiig it as: 2 ( ) ( ) ( )( ) ( ) P P P P P (9) k k k ( ) P if k is odd P k ( ) () if k is eve ad hece if, we get I if k is odd P k ( ) () if k is eve Let us coside some sets with followig otatios: { : is age symmetic} Γ (2) { : ( ) is age symmetic P } M M (3) ad { M : ( ) is age symmetic ad ( ) M P P } Λ (4) Theoem Let Γ, ad Λ be the sets defied i (2), (3) ad (4) espectively The, Γ ad Γ Λ M Poof The poof follows easily by utilizig Lemmas ad 3 Fom the statemet (i) of Lemma 3, it is obvious that P ( Γ ) Γ Howeve the 4 evese iclusio does ot hold i geeal Coside the matix If possible suppose thee exist a matix J M 2 ( ) Γ such that P ( J), the by Lemma 3, we have J, which is absud, sice 4 σ ( ) but 4 σ ( J ) Thee- foe Γ P ( Γ) emak 3 Let { }, the by usig Lemma P ( ) { } M M mp mp Λ { }, the by emak 2, we have P ( ) M { } ( Λ) { } ad if Λ, the by Theoem, we have P ( mp ) ( Λ) { } { I } The ext esult povides a chaacteizatio of the set { M M : P ( ) } Ψ Theoem 2 Let M ( ) { ( ) : mp } be age symmetic give by (2), the Ψ M M mp If 42

12 D Kishaswamy, M Loe D Poof Let The P ( ) Theefoe fo I 2, we have such that M I ie, M M M ad M the esult follows The ext esult shows that the fuctio P ( ), whe esticted to the set Γ is mootoically deceasig wt the patial odeig wt Mikowski adjoit Theoem 3 Let, Γ, such that The P ( ) P ( ) Poof Let, Γ, such that ice is age symmetic we have ad theefoe Also 2 2 (5) (6) Fially usig (5) ad (6) we get P ( ) P ( ) P ( ) ad P ( ) ( ) ( ) P P Hece P ( ) P ( ) 2 Howeve fo the age symmetic matices ad, we have P ( ) P ( ) but Thus we have the followig esult mp Theoem 4 Let, P P The, such that ( ) ( ) Poof The poof follows at oce by usig Theoem 3 ad emak 2 Theoem 5 Let be age symmetic ad M M be such that mp The P ( ) P ( ) P ( ) Poof Coside the decompositio of as give i (2) The fom Theoem 8 we D get M ad hece ( ) P P Thus if we assume ( M) P ( ) P ( ), the fom Theoem 6, we get ( ) P J with J I Theefoe J P ( M) Hece mp mp P ( ), Covesely if we assume that P ( ), the is age symmetic ad fially fom Theoem 3, the esult follows Theoem 6 Let M M has the epesetatio as give i poit ( v ) of Lemma ad M M The P ( ) P ( ) if ad oly if thee exists E E M M such that P P Poof Assume that P ( ) P ( ) The fom Theoem 3 we have P ( ) P ( ) P ( ) P ( ) P ( ) Let ( ) P P P, whee the blocks ae L M patitioed accodig to the blocks of sig the above metioed equality we have 43

13 D Kishaswamy, M Loe K, L ad M I ad hece P ( ) P P Also I I J I P E F ( ) P P Takig P P, whee the G H P The fom the equatio decompositio is doe accodig to the blocks of ( ) ( ) ( ) we get G, H, F, JE EJ ad JF E ad theefoe oe The covese is obvious Ackowledgemets P P with JE EJ Clealy E M ( ) has idex The secod autho was suppoted by GC-B though gat No F25-/24-5(B)/ 7-254/29(B) (225) This suppot is geatly appeciated efeeces [] Gohbeg, I, Lacaste, P ad odma, L (25) Idefiite Liea Algeba ad Applicatios Bikhäuse, Velag, Basel, Bosto, Beli [2] Meeakshi, A (2) Geealized Ivese of Matices i Mikowski pace Poceedigs of Natioal emia o Algeba ad Its Applicatios,, -4 [3] Meeakshi, A (2) age ymmetic Matices i Mikowski pace Bulleti of the Malaysia Mathematical cieces ociety, 23, [4] Meeakshi, A ad Kishaswamy, D (26) Poduct of age ymmetic Block Matices i Mikowski pace Bulleti of the Malaysia Mathematical cieces ociety, 29, [5] Loe, M ad Kishaswamy, D (26) m-pojectios Ivolvig Mikowski Ivese ad age ymmetic Popety i Mikowski pace Joual of Liea ad Topological Algeba [6] Kishaswamy, D (25) Cotibutios to the tudy o age ymmetic Matices i Mikowski pace PhD Dissetatio, Aamalai ivesity, Idia [7] Be-Iseal, A ad Geville, T (23) Geealized Ivese: Theoy ad Applicatios 2d Editio, pige Velag, New Yok [8] Campbell, L ad Meye J, CD (99) Geealized Ivese of Liea Tasfomatios 2d Editio, Dove, New Yok [9] Pasolov, VV (994) Poblems ad Theoems i Liea Algeba Ameica Mathematical ociety, Povidece [] Meye, CD (2) Matix Aalysis ad Applied Liea Algeba IAM, Philadelphia [] Mita, K, Bhimasakaam, P ad Malik, B (2) Matix Patial Odes, hoted Opeatos ad Applicatios Wold cietific Publishig Compay, igapoe [2] ao, C ad Mita, K (97) Geealized Ivese of Matices ad Its Applicatios Joh Wiley & os, New Yok [3] Tosic, M ad Cvetkovic-Ilic, D (22) Ivetibility of a Liea Combiatio of Two Matices ad Patial Odeigs Applied Mathematics ad Computatio, 28, [4] Malik, B (23) ome Moe Popeties of Coe Patial Ode Applied Mathematics ad 44

14 D Kishaswamy, M Loe Computatio, 22, [5] Malik, B, uedab, L ad Thome, N (24) Futhe Popeties o the Coe Patial Ode ad Othe Matix Patial Odes Liea Multiliea Algeba, 62, [6] Baksalay, JK ad Mita, K (99) Left-ta ad ight-ta Patial Odeigs Liea Algeba ad Its Applicatios, 49, [7] Deg, CY ad Wag, Q (22) O ome Chaacteizatios of the Patial Odeigs fo Bouded Opeatos Mathematical Iequalities & Applicatios, 5, [8] Liu, FX ad Yag, H (2) ome esults o the Patial Odeigs of Block Matices Joual of Iequalities ad Applicatios, 2, -7 [9] Mita, K (987) O Goup Iveses ad the hap Ode Liea Algeba ad Its Applicatios, 92, [2] Baksalay, JK, Hauke, J ad tya, GPH (994) O ome Distibutioal Popeties of Quadatic Foms i Nomal Vaiables ad o ome Associated Matix Patial Odeigs Multivaiate Aalysis ad Its Applicatios, 24, -2 [2] Baksalay, JK ad Putae, (99) Chaacteizatios of the Best Liea biased Estimato i the Geeal Gauss Makov Model with the se of Matix Patial Odeigs Liea Algeba ad Its Applicatios, 27, [22] Putae, ad tya, GPH (25) Best Liea biased Estimatio i Liea Models (Vesio 8) tatpob: The Ecyclopedia posoed by tatistics ad Pobability ocieties [23] tepiak, C (987) Odeig of Noegative Defiite Matices with Applicatio to Compaiso of Liea Models Liea Algeba ad Its Applicatios, 7, [24] Dazi, MP (978) Natual tuctues o emi Goups with Ivolutio Bulleti Ameica Mathematical ociety, 84, [25] Hatwig, E (98) How to Patially Ode egula Elemets? Japaese Joual of Mathematics, 25, -3 [26] Puithavalli, G (24) Cotibutios to the tudy o Vaious olutios of the Matix Equatio AXBC i Mikowski pace M PhD Dissetatio, Aamalai ivesity, Aamalai Naga [27] Headez, A, Lattazi, M, Thome, N ad quiza, F (22) The ta Patial Ode ad the Eigepojectio at o EP Matices Applied Mathematics ad Computatio, 28,

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