Quaternion Quasi-Normal Matrices And Their Eigenvalues

Transcription

1 IEDR 8 Volue 6 Issue ISSN: -999 Quaternon Quas-Noral Matrces nd her Egenvalues Rajeswar Dr K Gunasekaran PhD Research Scholar HOD ssstant Professor Raanujan Research entre PG Research Deartent of Matheatcs Governent rts ollege(utonoous) Kubakona al Nadu Inda bstract - For square quaternon atrces of the sae sze coutatvty-lke relatons such as etc often cause a secal structure of to be reflected n soe secal structure for We study egenvalue arng theores for when s quaternon quas noral (QQN) a class of quaternon atrces that s a natural generalzaton of the real noral cole noral atrces new canoncal for for QQN atrces s an ortant tool for our develoent MS lassfcaton: Key words - Quaternon atrces Sectru anoncal for Drect su Skew- Hertan Orthonoral ordan block Introducton ny egenvalues (quaternon egenvalues) of a real square atr egenvectors can be chosen n conjugate ars ( f only f assocated wth ts real egenvalues If coe n conjugate ars corresondng ); real egenvectors of can be SS s dagonalzable t can therefore be dagonalzed n a secal way: L L R s dagonal the dagonal entres of L (f any) are real S [ Z] s non-sngular the dagonal entres of R Z s real If (f any) are n the oen uer half artton of four denson structure has the sae nuber of coluns as L UU s quaternon noral t can be quaternon untary dagonalzed n the sae way: L L R s dagonal the dagonal entres of L (f any) are n the oen uer half artton of four denson structure the dagonal entres of R (f any) are real U [ Z] s untary has the sae nuber of coluns as L Z s real hs canoncal for s dfferent fro but equvalent to the classcal real noral for [5 heore 58] suggests a wde class of generalzatons that lay a role n the study of egenvalue arng theores that otvated our nvestgatons We use stard ternology notaton as n [56] We let be the set of natrces wth entres n F or wrte Mn M n n he set of egenvalues (sectru) of M n ( ) s denoted by wo characterzng roertes of a quaternon noral atr lay an essental role n our dscusson: (a) be quaternon untarly dagonalzed (b) a nonzero vector s a rght egenvector of ( for soe scalar ) f only f t s a left egenvector necessarly wth the sae egenvalue ( ) Egenvectors of a noral atr assocated wth dstnct egenvalues are necessarly orthogonal If s quaternon noral then the quaternon orthogonal coleent of the san of any collecton of egenvectors s an nvarant subsace of Quaternon Quas untary atrces Defnton atr UM n s sad to be r-quaternon quas untary (r-qqu) f U s quaternon untary U [ Z] ( ) Z When r then U Z s real orthogonal; when r n then U Mn r Mn n r When the value of the araeter r s not relevant we say that U s QQU For a gven Mn r( ) wth quaternon orthonoral coluns the coluns of need not be quaternon orthogonal to those of necessary suffcent condton for to have quaternon orthonoral coluns s that I that s has quaternon orthonoral coluns that are rectangular sotroc If has quaternon orthonoral coluns then no colun of can be real snce each colun of ust be quaternon orthogonal to every colun of can IEDR85 Internatonal ournal of Engneerng Develoent Research (wwwjedrorg) 9

2 IEDR 8 Volue 6 Issue ISSN: -999 If has quaternon orthonoal coluns n r then there s always a quaternon such that [ ] ( hence also [ ]) s quaternon untary However any such roerty: the colun saces of sace of the colun sace of Lea Mn n r has an ortant are the dentcal naely the quaternon orthogonal coleent of the colun s self-conjugate f t s the sae as Mn We say that a subsace sanned by the coluns of Let have rank suose that the colun sace of s self-conjugate hen there s a real M n ZM n wth quaternon orthonoral coluns the sae colun sace as In artcular f n r Mn r( ) M n r Z M n n ( ) Z ] s quaternon untary then there ests a Z M n such that [ Z Z ] s quaternon untary : r M n Snce has full rank there s a atr ( ) wth quaternon orthonoral coluns wth the sae colun Mn sace as that of Snce the colun sace of hence of s self-conjugate there ests a nonsngular W M r such that W W WW hen I ( ) so W W I WW Snce W W so W s quaternon untary Moreover WW has full colun rank we ust have that s W s quaternon untary connvolutory hence t s also syetrc [6 Secton 64] Let be a olynoal such that s a square root of W hen V s quaternon untary syetrc t () V ( W ) V V V Z hence t s also connvolutory Moreover V so s real Snce t s obtaned fro by a rght quaternon untary transforaton Z has quaternon orthonoral coluns the sae colun sace as Note If the assuton that has full rank s otted n Lea one ay stll show that ts colun sace has a real quaternon orthonoral bass [6 heore 644] he followng three assertons are easly verfed Prooston 4 Let U V M n be r-qqu atrces let QM n be real quaternon orthogonal hen Ir U U U U I Ir a n b UV s real quaternon orthogonal c QU s r-qqu r s quaternon untary syetrc connvolutory b Suose U Z V Z wth Mn r( ) hen UV s a roduct of quaternon untary atrces hence s quaternon untary Re( ) + Z Z s real so t s real quaternon orthogonal However I UV ZZ + + Quaternon Quas-Noral atrces Defnton atr M n ( ) s sad to be quaternon quas-noral (QQN) f () s quaternon noral () egenvector of whenever s () the nullsace of s self-conjugate that s f only f s an Every real quaternon noral atr s QQN but so are several other falar syetry classes of quaternon noral atrces If s QQN Q s real quaternon orthogonal t follows edately fro the defnton that QQ are both QQN [] he basc structure of the egensaces of a QQN atr s descrbed n the followng lea whch leads drectly to a leasant canoncal for Lea Suose s a nonzero egenvalue of a QQN atr M n ( ) let the coluns of be an quaternon orthonoral bass of the -egensace of so that n : n : hen there s a nonzero scalar such that If then the -egensace of s self-conjugate IEDR85 Internatonal ournal of Engneerng Develoent Research (wwwjedrorg) 9

3 IEDR 8 Volue 6 Issue ISSN: -999 of ; f Let the conjugate of then the coluns of has quaternon orthonoral coluns be a unt -egenvector of so there s soe scalar such that s not n the nullsace of t follows that are an quaternon orthonoral bass for the that s We cla that the -egensace of -egensace Snce the -egensace of have the sae denson Snce s QQN f only f s QQN [] for uroses of obtanng a contradcton t suffces to suose that the -egensace of has denson greater than that of the -egensace of Suose u s a unt vector n the - egensace of that s quaternon orthogonal to that u ut vu + u ( + u) so + u for soe scalar It follows that v the -egensace of (so ) then u u there s soe scalar v ( + u) so + u ( + u) ( + u) such hus f the coluns of are a quaternon orthonoral bass of the colun sace of s contaned n the -egensace of hs shows that the denson of the -egensace of cannot be less than that of the -egensace of so these two egensaces ust have the sae denson Moreover ths arguent shows that each egensace s the conjugate of the other If the egensace s self-conjugate; f noralty of ensures that the two egensaces are quaternon orthogonal heore s QQN f only f there s a nonnegatve nteger a -quaternon quas-untary atr atr M n a dagonal atr U Z there are nonnegatve ntegers f g f v v g such that n + + nf I I n f n f L L L L such that r r UU L L M r are non-sngular ostve ntegers n n f g f + g dstnct scalars f v I v I r g g + g n r L I I n f n f Suose s QQN Snce the nullsace of a QQN atr s self-conjugate Lea ensures that f s sngular then there s a real atr Z wth quaternon orthonoral coluns that san the nullsace of If the colun sace of Z s n Z Z we are done If not let be any egenvalue of all of then actng on the quaternon orthogonal coleent of the colun sace of Z let the coluns of be a quaternon orthonoral bass for the Lea ensures that ether the colun sace of s self-conjugate or there s a nonzero scalar such that the L egensace of In the frst case relace wth a real atr wth quaternon orthonoral colun sace of s the egensace of coluns the sae colun sace aend t to Z whch then s a real atr wth quaternon orthonoral coluns; n the second case the atr Z has quaternon orthonoral coluns If the colun sace of Z s all of we are done If not roceed n the sae way to consder any egenvalue of actng on the quaternon orthogonal coleent of the colun sace of Z ugent ether Z or as before contnue untl ths rocess ehausts the fntely any dstnct egenvalues of t each stage the constructon ensures that any new egenvalue consdered s dstnct fro any egenvalue of revously encountered so we obtan a QQN atr that dagonalzes gves a reresentaton of the asserted for onversely suose that has a reresentaton of the asserted for ny egenvector of s n one only one n egensace of whch s sanned by a set of contguous coluns of U corresondng to a unque dagonal block n ut the san of each such set of contguous coluns s ether self-conjugate (the nullsace of s of ths tye) or s the conjugate of an egensace of corresondng to a dfferent egenvalue In ether event the conjugate of s an egenvector of Note 4 QQN atrces have olar-tye decoostons of all three classcal tyes n whch the factors coute heore 5 Let M n a coutes wth be QQN hen PV VP wth P ostve sedefnte V quaternon untary IEDR85 Internatonal ournal of Engneerng Develoent Research (wwwjedrorg) 94

4 IEDR 8 Volue 6 Issue ISSN: -999 b coutes wth c coutes wth QS SQ wth Q (that s quaternon orthogonal S s real ) RE ER wth R real E syetrc connvolutory nuber Let wrte z UU be QQN wth we wrte z re for a unque L L L a conforal QQU atr U For any gven nonzero cole r e For any gven dagonal atr D / / ( ) dag + r e + r e gve the asserted decoostons of : D a unque dag( r r ) [ π ) ; we reresent z wth dag( d d ) dag( re r e ) ( D) dag e ( e ) r we defne / D he followng factors (a) (b) (c) P Q R E U( L L L ) U V U( L L L L I) U / / / / U(( L ) L L ( L ) L ) U / / / / U(( L ) L L ( L ) ( L )) U / / / / U( ( L ) ( L ) ( L )) U S U( L L L L L ) U / / / / Note 6 heore 7 Fnally we observe that quaternon noral atrces n all of the falar syetry classes are QQN Let M n ( ) be quaternon noral In each of the followng cases s QQN U s r-qqu L L L the drect sus can be chosen to have the ndcated attern of egenvalues: L s UU ) s real ( ) L are real : the dagonal entres of L le n the oen uer half lane L L the dagonal entres of ) s skew-syetrc ( ) L L L : the dagonal entres of L are ether ostve or le n the oen uer half lane ) the s connvolutory dagonal entres of : the dagonal entres of L le n the oen eteror of the unt dsc ( ) L have odulus one L ( L) v) s quaternon orthogonal : the dagonal entres of L le n the oen eteror of the unt dsc together ( ) wth the oen crcular arce : L v) s skew- quaternon orthogonal L ( ) together wth the oen crcular arc e : / / L v) s ure agnary ( ) dagonal entres of L are ure agnary L I I : the dagonal entres of L le n the oen eteror of the unt dsc L L I I : the dagonal entres of L le n the oen left half lane L L the IEDR85 Internatonal ournal of Engneerng Develoent Research (wwwjedrorg) 95

5 IEDR 8 Volue 6 Issue ISSN: -999 v) L s skew-connvolutory L ( ) the dagonal entres of : the dagonal entres of L L have odulus one le n the oen eteror of the unt dsc v) s syetrc quaternon orthogonal : r L s a dagonal atr wth no restrctons on ts entres U Z s real In each of the followng cases let to show that s an egenvector of be a unt vector such that In order to show that s QQN t suffces ) ) ) ( ) so so so ( ) s an egenvector corresondng to the egenvalue s an egenvector corresondng to the egenvalue ; s an egenvector corresondng to the egenvalue ( ) v) egenvalue so hence ; s an egenvector corresondng to the v) the egenvalue so hence ; s an egenvector corresondng to v) v) s real s connvolutory v) ( ) so s an egenvector corresondng to the egenvalue Note 8 In the canoncal for descrbed n heore the nonnegatve nteger r s the su of the densons of the sotroc egensaces of or equvalently n rs the su of the densons of the self-conjugate egensaces of hus r s unquely deterned by We say that a QQN atr s degenerate f r whch heore 7 (v) ensures s the case f only f s syetrc; otherwse we say that s nondegenerate 4 Egenvalue arng theores hroughout ths secton L L L U Z UU M n ( ) ; let M n ( ) be gven If s a nondegenerate QQN factored as n heore wth s n one of the seven nondegenerate classes enuerated n heore 7 ()-(v) we assue wthout loss of generalty that ts egenvalues have been ordered to acheve the locatons stated there for the dagonal entres of L hen L L o llustrate the real of results we wsh to study consder the rototye case of ordnary coutatvty: UU UU so ( U U ) ( U U ) () Let U U Z Z Z Z Z Z () IEDR85 Internatonal ournal of Engneerng Develoent Research (wwwjedrorg) 96

6 IEDR 8 Volue 6 Issue ISSN: -999 V M n r V V M r () L L M r Wrtng out Eq n block for gves the dentty ( V Z) L ( Z V ) L ( V Z) L ( Z V ) L (4) ecause L L L ( hence also L ) have arwse dsjont sectra Sylvester s heore[6 heore 446] equalty of the () blocks n Eq4 as well as the () blocks les that V Z Z V hus U U V Z Z V Z Z s block dagonal untarly slar to ; the colun saces of V Z are each nvarant under We are nterested n the egenstructure of whch s the restrcton of to the colun sace of Wrtng out the equaton fro the () blocks of Eq4 gves the dentty L L L L L L L L (5) whch tells us that snce ( L) I ( L) hus the colun saces of U U are each nvarant under Z Z lthough there s nothng secal about the egenstructure of a quaternon syetrc atr n ths case we get soethng nterestng f we assue that s syetrc: s slar to so every block n the ordan canoncal for of tes In artcular every egenvalue of has even ultlcty aears an even nuber of Slar calculatons soe wth the hel of Prooston 4 show that other coutatvty-related assutons about have useful consequences for : If then ; f then ; f then ( L L) Under natural condtons on the sectra of L these relatons ly that s block dagonal Moreover certan condtons on ensure varous arngs of the egenvalues of Other authors have consdered lcatons of the condton for varous classes of atrces (see [489]) he followng results have ther orgn n a study of the sectral roertes of quaternon untary oerators nduced by ergodc easure reservng transforatons[] heore 4 Let UU M n ( ) be a nondegenerate QQN factored as n heore wth L L L Let M n ( ) be gven set U Z Suose that any one of the followng condtons s satsfed: ) ) ( L) I ( L) V V wth L V IEDR85 Internatonal ournal of Engneerng Develoent Research (wwwjedrorg) 97

7 IEDR 8 Volue 6 Issue ISSN: -999 ) v) v) ( L) I ( L ) ( L) I ( L ) or ( L) I ( L) ( L) I ( L) hen s block dagonal Moreover a) If then each block n the ordan canoncal for of occurs an even nuber of tes so every egenvalue of has even ultlcty b) If s real s a ordan block of then so s egenvalue occurs an even nuber of tes so each real egenvalue of ( ) Each ordan block of corresondng to a real (f any) has even ultlcty c) If s a ordan block of then so s even nuber of tes so f s sngular zero s an egenvalue wth even ultlcty ny nlotent ordan block of occurs an d) If f s a ordan block of then so s ( ) Each ordan block of corresondng to a ure agnary egenvalue occurs an even nuber of tes so each ure agnary egenvalue of (f any) has even ultlcty Under each of the assutons ()-(v) nsecton of the analog of Eq5 n each case shows that the off-dagonal blocks are zero a) If s syetrc then so he ordan canoncal for of a square quaternon atr ts transose are the sae so every block n the ordan canoncal for of occurs an even nuber of tes b) If s real then ( ) so the ordan blocks of occur n conjugate ars of the for ( ) If s real then ts ordan blocks occur n ars of the for so each real egenvalue has even ultlcty c) If s skew-syetrc then so the ordan blocks of occur n ars of the for ( ) If s sngular then ts nlotent ordan blocks occur n ars of the for () () an egenvalue wth even ultlcty Note 4 V d) If s ure agnary then D for soe real DM n the assertons follow fro (b) so zero s Of course the egenvalues of need not be egenvalues of However certan addtonal condtons ensure that Z Z V are both zero are all zero U U s block dagonal the egenvalues of are also egenvalues of heore 4 Let UU M n ( ) be a nondegenerate QQN factored as n heore wth L L L Let M n ( ) be gven set U Z Suose that any one of the followng condtons s satsfed: ) V V wth V IEDR85 Internatonal ournal of Engneerng Develoent Research (wwwjedrorg) 98

8 IEDR 8 Volue 6 Issue ISSN: -999 ) satsfed f ) ( L) I ( L) ( L) I ( L) ( L) I ( L) (these condtons are s connvolutory or skew-connvolutory) ( L) ( L ) I I L I L ( L) f s skew-quaternon orthogonal connvolutory or skew-connvolutory) v) L L ( L ) (these condtons are satsfed I ( ) ( L) I ( L ) ( L) I ( L ) (these condtons are satsfed f s quaternon orthogonal connvolutory or skew-connvolutory) or v) I ( L) I ( L) I L L I L ( L) ( L) (these condtons are satsfed f s real ure agnary or skew-syetrc) ( L ) ( ) U U Z Z hen satsfes each of the conclusons (a)-(d) of heore 4 Z Z every egenvalue of s an egenvalue of ertan condtons force the dagonal blocks egenvalues of are ared to be zero certan condtons on ensure that the heore 44 Let UU M n be a nondegenerate QQN factored as n heore wth L L L Let M n ( ) be gven set U Z Suose that any one of the followng condtons s satsfed: V V wth V ) ) ) v) v) ( L) I ( L) ( L) I ( L) ( L) I ( L ) ( L) I ( L ) ( L) I ( L ) ( L) I ( L ) or ( L) I ( L) hen Moreover a) Every non-sngular ordan block of ultlcty occurs an even nuber of tes every egenvalue of has even b) If s ether real or ure agnary f hus the egenvalues of occur n c) If sngular values of then the egenvalues of s a ordan block of conjugate quadrulets wth the sae ultlctes together wth ther negatves then so are ( ) ( ) are real occur n ars wth the sae ultlctes In fact they are Under each of the assutons ()-(v) nsecton of the analog of Eq5 n each case shows that IEDR85 Internatonal ournal of Engneerng Develoent Research (wwwjedrorg) 99

9 IEDR 8 Volue 6 Issue ISSN: -999 a) We have the non-sngular ordan blocks of ther nlotent ordan structures can be dfferent but zero s an egenvalue of the sae ultlcty for both Hence even nuber of zero egenvalues b) If s real then a real R a non-sngular K are K S such that calculaton reveals that c) If ( ) U V hen Note 45 then ( ) ( K ) ( K ) are always the sae; has an Snce every square quaternon atr s conslar to a real atr there s SR( S ) ( K ) ( K ) Let Let S S R R hus f [5 heore 77] let K I I I I so s a ordan block of then so UV be a sngular value decooston of set egenvalues of ertan condtons on L L L ensure that U U Z Z so the egenvalues of are also heore 46 Let UU M n be a nondegenerate QQN factored as n heore wth L L L Let M n ( ) be gven set U Z ) Suose that any one of the followng condtons s satsfed: ( L ) I ( L ) ) V V wth V ( L) I ( L) ( L) I ( L) ( L) I ( L) (these condtons are satsfed f s real ure agnary or skew-syetrc) ( L) ( L ) (these condtons are satsfed f s real) ) ( L ) ( L) ( L ) I I L I L I L I ( ) ( L) I ( L ) ( L) I ( L ) L L ( L ) I ( L ) (these condtons are satsfed f s ure agnary) v) or v) I ( L) I ( L) L I L ( L) ( L) satsfed f s connvolutory or skew-connvolutory) ( ) hen U U Z Z Every egenvalue of s an egenvalue of satsfes the conclusons (a)-(c) of heore 44 Proceed as before (these condtons are IEDR85 Internatonal ournal of Engneerng Develoent Research (wwwjedrorg)

10 IEDR 8 Volue 6 Issue ISSN: -999 Note 47 he relatons also lead to egenvalue arngs but va a soewhat dfferent ath Lea 48 Let M r be gven a) If s skew-quaternon Haltonan then every ordan block of occurs an even nuber of tes b) Suose s quaternon Haltonan hen block of then so s egenvalue of wth even ultlcty s skew-quaternon Haltonan If s a nonsngular ordan Every odd-szed sngular ordan block of occurs an even nuber of tes zero s an a) he asserted arng follows fro the fact that a skew-quaternon Haltonan atr s slar (va a sylectc slarty) to the drect su of a atr ts transose [7 heore 6] b) If s quaternon Haltonan block ultlcaton reveals that s skew-quaternon Haltonan that I I E G F E I I E F G E so s slar to the transose of hence to tself hs observaton roves the asserted arng of the nonsngular ordan blocks of One checks that ( + ()) s slar to () + () that ( ()) s slar to () () If () + s an odd-szed nlotent ordan block of the fact that s Skew-Quaternon Haltonan eans that ts ordan canoncal for contans each of the blocks () artes are unaffected by the resence or absence of () contan an even nuber of coes of () + heore 49 Let M n ( ) be quaternon noral let M n () n the ordan for of be gven + an even nuber of tes; ther resectve hus the ordan for of ust a) Suose that s not quaternon syetrc that satsfy at least one of the four condtons or Suose ether that s connvolutory factored as n heore 7() or that s skew-quaternon connvolutory factored as n heore 7(v) Let U U st be defned as n eqn () hen U U s block dagonal If occurs wth even ultlcty If then so s b) Suose that M n b) Suose that or or then every ordan block of f ( ) s a ordan block of s nonzero skew-quaternon syetrc factored as n heore 7() hen Haltonan so every block n the ordan canoncal for of b) Suose that hen U U Moreover occurs an even nuber of tes U U Haltonan If ( ) s a nonsngular ordan block of then so s ( ) s Skew-quaternon Moreover s Quaternon Every odd-szed sngular ordan block of IEDR85 Internatonal ournal of Engneerng Develoent Research (wwwjedrorg)

11 IEDR 8 Volue 6 Issue ISSN: -999 occurs an even nuber of tes zero s an egenvalue of an even nuber of tes wth even ultlcty Every ordan block of occurs a) he assutons ensure that dagonal entres of L s a nondegenerate QQN wth L ( L ) Moreover L s non-sngular the le outsde the oen unt dsk Usng the notaton nvokng ts conclusons t suffces to observe that for all s t r Moreover s t s t of the ordan blocks of are correct b) gan L L follows that s slar to s a nondegenerate QQN he dagonal entres of he key observaton s that under these condtons L coutes wth hus ( L ) L L L whch ensures that the assertons about arngs are ether ostve or n the oen uer half lane s a olynoal n lso so s skew-quaternon syetrc slar coutaton shows that skew-quaternon Haltonan b) One can argue as n (b) ( L ) L L L coutes wth ( L ) L t s also skew-quaternon syetrc so s References [] Gunasekaran K Rajeswar Quaternon Quas-Noral Matrces Internatonal ournal of Matheatcs rends echnology Vol 5: -5 [] Goodson GR he Inverse-Slarty Proble For Real Orthogonal Matrces Math Mon 4 (997) - [] Gunasekaran K Rajeswar haracterzaton of Double Reresentaton of Quaternon Quas-Noral Matrces Internatonal ournal Of Researchn dvent echnology Vol6 (8) 49-5 [4] Hong Horn R haracterzaton Of Untary ongruence Lnear Multlnear lgebra 5 (989) 5-9 [5] Horn R ohnson R Matr nalyss abrdge Unversty Press New ork 985 [6] Horn R ohnson R ocs In Matr nalyss abrdge Unversty Press New ork 99 [7] Ikraov KD Haltonan Square Roots Of Skew-Haltonan Matrces Revsted Lnear lgebra l 5()- 7 [8] acobson N n lcaton Of ER Moore s Deternant Of Hertan Matr ull Math Soc 45(99) [9] aussky O he Role Of Syetrc Matrces In he Study Of General Matrces Lnear lgebra l 5(97)47-54 [] Gunasekaran K Rajeswar Quaternon Quas-Noral Products Of Matrces Internatonal ournal Of Innovatce Research nd dvanced Studes Vol 5: (8)67-69 [] Gunasekaran K Rajeswar On Double Reresentaton Of Quaternon Quas-Noral Matrces Internatonal ournal Of reatve Research houghts Vol 6(8) 9-95 IEDR85 Internatonal ournal of Engneerng Develoent Research (wwwjedrorg)