Generalized k-normal Matrices

Transcription

1 Iteatioal Joual of Computatioal Sciece ad Mathematics ISSN Volume 3, Numbe 4 (0), pp 4-40 Iteatioal Reseach Publicatio House Geealized k-omal Matices S Kishamoothy ad R Subash Depatmet of Mathematics, Govemet Ats College (Autoomous), Kumbakoam, Tamiladu, Idia 6 00 Depatmet of Mathematics, AVC College of Egieeig, Maampadal, Mayiladuthuai, Tamiladu, Idia subash_u@ediffmailcom Abstact A matix A is called geealized k-omal povided that thee is a positive defiite k-hemite matix H such that HAH is k-omal I this pape, these matices ae ivestigated ad thei caoical fom, ivaiats ad elative popeties i the sese of coguece ae obtaied AMS classificatio: 5A04, 5A8 Keywods: Geealized k-omal matix, coguece, caoical fom ad ivaiat Itoductio Let be the space of x complex matices Let k be a fixed poduct of disoit taspositio i S ={,, } (hece, ivolutay) ad let K be the associated pemutatio matix of k Two matices AB, ae called k-uitaly simila povided that thee exists a k-uitaly matix U such that B = KU AU, whee U is the cougate taspose matix of U Fo a matix A, thee exists a k-uitay matix U such that KU AU is a uppe tiagula matix ad A is k-uitaly simila to a k-diagoal matix if ad oly if A is k-omal Two matices AB, ae coguet if thee exists a osigula matix P such that B = KP AP It is easy to see that, fo a give matix A, thee exists a osigula matix

2 4 S Kishamoothy ad R Subash P such that KP AP is a uppe tiagle matix sice k-uitay simila matices A, B ae coguet I this pape, we obtai the caoical fom ad ivaiats fo geealized k- omal matices i the sese of coguece We give some detemiatal iequalities fo geealized k-omal matices ad thei k-hemite pat ad skew k-hemite pat Basic Defiitios Defiitio : A matix A is said to be k-hemitia, if A= K A K That is, a = a ( ) ( ), i,=,,, i k k i Example : 0 0 A= 0 0 i i 0 0 is k-hemitia Defiitio 3: A matix A is said to be skew k-hemitia, if A= K A K That is, ai = ak( ) k( i), i,=,,, Example 4: 0 i 0 A= i i is skew k-hemitia Geealized k-omal matices Defiitio 3[4]: A matix A is said to be k-omal, if AA K = K A A That is, aia k( ) + k( i) = ak( ) k( i) + ai; i, =, Example 3: 0 i 3i A= i 3i 0 3i 0 i is k-omal Defiitio 33[4]: A matix A is said to be k-uitay, if AA K = K A A= K Example 34: i A = is k-uitay i

3 Geealized k-omal Matices 43 Defiitio 35: A matix A is called geealized k-omal if thee is a positive defiite k-hemite matix H such that HAH is k-omal i Example 36: H = + i is a positive defiite k-hemite matix i The A = 0 i is geealized k-omal Theoem 37: A is geealized k-omal if ad oly if thee exists a positive defiite k-hemite matix X such that KAXA= AXAK Poof: Assume that A is geealized k-omal matix The by defiitio (35), thee exists a positive defiite k-hemite matix H such that HAH is k-omal Theefoe, K( H AH) ( H AH) = ( H AH)( HAH ) K KH AH HAH= HAHH AH K K( KHK) A( KHK) HAH= HAH( KHK) A( KHK) K HKA( KH) AH= HA( HK) AKH KA ( KH) A= A( HK) AK () Sice H is a positive defiite, H is o-sigula The H is positive defiite Let X = H = H K = ( H K ) = ( KH) The X is a positive defiite k-hemite matix Thee foe () KAXA= AXAK Covesely, Assume that KAXA= AXAK ad X is a positive defiite k- hemite matix The thee exists a positive defiite k-hemite matix H such that X = H = H K = ( H K ) = ( KH) Thus KA( KH) A= A( HK) AK HKAKHKHAH= HAHKHKAKH ( KH K) K A H H AH = H AH H A K( KH K) KH AH HAH= HAHH AH K K( H AH) ( H AH) = ( H AH)( H AH) K Hece HAH is k-omal ad the A is geealized k-omal Theoem 38: A is geealized k-omal if ad oly if A is coguet to a k- diagoal matix

4 44 S Kishamoothy ad R Subash Poof: Assume that A is geealized k-omal matix The by defiitio (35), thee exists a positive defiite k-hemite matix H such that HAH is k-omal By Schu s theoem [], thee exists a k-uitay matix U such that KU ( H AH) U is k-diagoal matix Let P = HU P = UH = UKHK P H= U KHKH P H U ( KH) = osigula ad KP AP matix P H = U H K P H = U H P = UH The P is is k-diagoal matix That is, A is coguet to a k-diagoal Covesely, Assume that A is coguet to a k-diagoal matix The thee exists a osigula matix P such that KP AP= Dis a k-diagoal matix By the pola factoizatio theoem of matices, thee exists a positive defiite k-hemite matix H ad a k-uitay matix U such that P = HU P = UH = UKHK P H= U KHKH P H = U ( KH) P H = U H K P H = U H P = UH Thus KU ( H AH) U = Dis a k-diagoal matix By Schu s theoem [] ad defiitio (35), A is geealized k-omal Theoem 39: Let A be a geealized k-omal matix The thee exists a iθ iθ iθ osigula matix P such that A = KP P, whee D = diag( e, e,, e ) iθ iθ iθ, = ak(a), π θ θ θ < π, θ θ < π ad e, e,, e ozeo k-eige values of P ( KP ) A ae all Poof: Sice A is geealized k-omal matix By Theoem (38) thee exists a osigula matix Q such that A= KQ diag( λ, λ,, λ,0,0,,0) Q, whee = ak(a) is the ak of A, λ, λ,, λ ae ozeo complex umbes iθ Let λ = ρ e, ρ > 0, =,,, ad π θ θ θ < π Deote R = diag( ρ, ρ,, ρ,,,) ad P = RQ P = Q R The P is osigula ad A = KP P 0 0 D D Sice P ( KP ) A = P ( KP ) ( K P ) P P iθ iθ iθ = P, e, e,, e 0 0 ae all ozeo k-eige values of ( ) P KP A

5 Geealized k-omal Matices 45 Without loss of geeality, π θ θ θ < π(mod π) The θ θ θ θ < π(mod π) Thus θ θ < π Sice A = KP P, D 0 0 D P ( KP ) A = P ( KP ) ( K P ) P = P P iθ iθ iθ Hece e, e,, e ae all ozeo k-eige values of P ( KP ) A Theoem 30: Let A be a geealized k-omal matix If thee exist a osigula matix P ad Q such that A K P P KQ = = Q, whee iθ iθ iθ D= diag( e, e,, e ), π θ θ θ < π, θ θ < π, = ak(a), iw iw iw D = diag( e, e,, e ), π w w w < π ad w w < π The θ = w fo Poof: Give A K P P KQ = = Q K P P KQ = Q Pe multiply by K o both sides Theefoe, P P Q = Q D 0 D 0 = ( P ) Q QP R R Deote QP = R=, whee R R R is a x complex matix R R ( QP ) = ( P ) Q = R = R R () R R R R D 0 = D 0 D 0 R 0 0 R R R RD 0 R R RDR RDR RD 0 R R RDR RDR = = ()

6 46 S Kishamoothy ad R Subash Thus D R DR = Sice D is osigula, Rad R ae also osigula Hece D = D( D ) = ( RDR )( R D ( R) ) = RD ( D ) ( R) = RD ( R) That is, D ad D ae simila Thus D ad D have the same k-eige values iθ That is, e iw = e fo Theefoe, θ w (mod π ) fo ad hece θ w (mod π ) fo By the coditios of the theoem, θ = w fo Remak 3: Theoem (30) idicates that whe a geealized k-omal matix A iθ iθ iθ satisfies A = KP P, the D= diag( e, e,, e ) is idepedet of the choice of the osigula matix P, that is, the k-diagoal matix D is oly detemied by A So we have the followig defiitios Defiitio 3: Let A be a geealized k-omal matix satisfyig iθ iθ iθ A = KP PDeote DKA ( ) = D= diage (, e,, e ), whee = D 0 ak(ka), π θ θ θ < π ad θ θ < π The is called the caoical fom of the geealized k-omal matix A i the sese of coguece ad iθ iθ iθ k ( A) ( e, e,, e ) σ = is called the geealized k-spectum of A Theoem 33: Let AB, be geealized k-omal matices The A ad B ae coguet if ad oly if σ k( A) = σ k( B), that is A ad B have the same geealized k- spectum Poof: Assume that A ad B ae coguet The thee exists a osigula matix Q such that B = KQ AQ By Theoem (39), thee exists a osigula matix P such that DA ( ) 0 A = KP P 0 0 DA ( ) 0 Thus B KQ K P = PQ (3) 0 0 Deote R = PQ, the R is osigula R = PKQK R = KQ KP

7 Geealized k-omal Matices 47 DA ( ) 0 DKA ( ) 0 Theefoe (3) B = R R KB = R R By a Theoem (30), D(KB) = D(KA) Hece σ ( KA ) = σ ( KB ) σ k( A) = σ k( B ) Covesely, Let σ k( A) = σ k( B ) σ ( KA ) = σ ( KB ) D(KA) = D(KB) = D ad thee exists a osigula matices P, Q such that A = KP P (4) ad B = KQ Q (5) Pe multiply by ( KP ) ad post multiply by P i (4) Theefoe (4) ( KP ) AP = Hece B = ( KQ )( KP ) AP Q Theefoe A ad B ae coguet Detemiatal popeties Let A Alteatively, let H ( A) = ( A+ KA K), the k-hemite pat ad S( A) = ( A KA K), the skew k-hemite pate It is kow that A= H( A) + S( A) ad KAK = H( A) S( A) A is called positive defiite povided that H(A) is positive defiite k-hemite matix It is easy to kow that A is positive defiite if ad oly if Re( Kx, Ax) > 0 fo each ozeo complex colum vecto x = ( x, x,, x ) T, whee Re(a) deotes the eal pat of a complex umbe a Theoem 4: Let A be a positive defiite matix, the A is geealized k- omal Poof: sice A is positive defiite, H(A) is osigula The KA K[ H( A)] A= [ H( A) S( A)][ H( A)] [ H( A) + S( A)] { H( A)[ H( A)] = S( A)[ H( A)] }[ H( A) + S( A)] { I S( A)[ H( A)] }[ H( A) S( A)] = + = HA+ SA SA HA HA+ SA ( ) ( ) ( )[ ( )] [ ( ) ( )] = H A + S A S A H A H A S A H A S A ( ) ( ) ( )[ ( )] ( ) ( )[ ( )] ( )

8 48 S Kishamoothy ad R Subash = HA+ SA SA SA HA SA ( ) ( ) ( ) ( )[ ( )] ( ) = H( A) S( A)[ H( A)] S( A) (5) A H A KA K H A S A H A H A S A [ ( )] = [ ( ) + ( )][ ( )] [ ( ) ( )] { I S( A)[ H( A)] }[ H( A) S( A)] = + = H A S A + S A H A H A S A ( ) ( ) ( )[ ( )] [ ( ) ( )] = HA SA+ SA SA HA SA ( ) ( ) ( ) ( )[ ( )] ( ) = H( A) S( A)[ H( A)] S( A) (6) Fom (5) ad (6), we get KA K [ H( A)] A AH [ ( A)] = KAK (7) Sice H is a positive defiite k-hemite matix Hece [ H( A)] is a positive defiite k-hemite matix By a Theoem (37), X K[ H( A)] = = [ H( A)] Kis a positive defiite k-hemite matix Theefoe (7) KA XA = AXA K Hece A is geealized k-omal matix Theoem 4: Let be a itege with 3 ad A a geealized k-omal matix The det ( A) det H( A) + det S( A) Futhe, the equality holds if ad oly if = ak(a)< o A is a k-hemite matix o A is a skew k-hemite matix Poof: Thee exists a osigula matix P such that A = KP P It is easy to see that HA ( ) = KPHDP ( ) ad SA ( ) = KPSDP ( ) Thus ak [H(A)]=ak [H(D)] ad ak [S(A)]= ak[s(d)] We distiguish the followig cases Case (): If ak (A)< Tha ak [H(A)]=ak [H(D)] ak (A)< ad ak [S(A)]= ak[s(d)] ak (A)< Thus det ( A) = det H( A) = det S( A) = 0 ad so det ( A) = det H( A) + det S( A) iθ iθ iθ Case (): If ak (A)= Now, D = diag( e, e,, e ), D = diag(cosθ + isi θ,cosθ + isi θ,,cosθ + isi θ ) H ( D ) = diag(cos θ,cos θ,,cos θ ), i A= KP DP The det( A) (det KP) e θ = Π = = det S( A) = i (det KP) Π siθ = S( D) = idiag(si θ,si θ,,si θ ) ad det H( A) = (det KP) Πcosθ ad

9 Geealized k-omal Matices 49 Sub case (): If cosθ = 0 fo some = The det H ( A) + det S( A) = det KP Π siθ det KP det ( A) Futhe, the above equality holds if ad oly if siθ = fo all, if ad oly if cosθ = 0 fo all, if ad oly if H(A)=0 if ad oly if A is skew k-hemitia Sub case (): If siθ = 0fo some The θ = det H ( A) + det S( A) = det KP Π cos det KP det ( A) Futhe, the above equality holds if ad oly if cosθ = fo all, if ad oly if siθ = 0fo all, if ad oly if S(A)=0 if ad oly if A is k-hemitia Sub case (3): If cosθ 0 ad siθ 0fo all Now det H( A) + det S( A) = det KP Π cosθ + Π siθ = = det HA ( ) + det SA ( ) = det KP ( cosθcos θ + siθsi θ ) det KP = det ( A) By summig up the above cases the poof of this theoem is completed Remak 43: Let x, x,, x, y, y, y be o egative eal umbes The ( x, x,, x ) + ( y, y, y ) ( x + y ) ( x + y ) ( x + y ) ad the equality holds if ad oly if thee exists ad itege such that x = y = 0 o ( x, x,, x ) ad ( y, y, y ) ae liealy depedet Theoem 44: Let be a itege with 3 ad A a geealized k-omal iθ iθ iθ matix with geealized k-spectum σ ( A) = ( e, e,, e ), whee = ak(a) k The det ( A) det H( A) + det S( A) Futhe, the equality holds if ad oly if < o (cos θ,cos θ,,cos θ ) ad (si θ,si θ,,si θ ) ae liealy depedet Poof: Thee exists a osigula matix P such that A = KP P It is easy to see that HA ( ) = KPHDP ( ) ad SA ( ) = KPSDP ( ) Thus ak [H(A)]=ak [H(D)] ad ak [S(A)]= ak[s(d)] We distiguish the followig cases Case (): If ak (A)< Tha ak [H(A)]=ak [H(D)] ak (A)< ad ak [S(A)]= ak[s(d)] ak

10 40 S Kishamoothy ad R Subash (A)< Thus det ( A) = det H( A) = det S( A) = 0 Thee foe det ( A) det H( A) + det S( A) iθ iθ iθ Case (): If ak (A)= Now, D = diag( e, e,, e ), H ( D ) = diag(cos θ,cos θ,,cos θ ), S( D) = idiag(si θ,si θ,,si θ ) ad i A= KP DP The det ( A) (det KP) e θ = Π det H( A) = (det KP) Πcosθ ad det ( ) (det ) si = S A = i KP Π θ Thus det H( A) + det S( A) = det KP Π cosθ + Πsiθ = = 4 det H( A) + det S( A) = det KP Π (cos θ) +Π(si θ ) = = = By Remak (43), det H ( A) + det S( A) det ( A) ad the equality holds if ad oly if thee exists a itege such that cos θ = si θ = 0 o (cos θ,cos θ,,cos θ ) ad (si θ,si θ,,si θ ) ae liealy depedet Sice cos θ + si θ =, the equality holds if ad oly if (cos θ,cos θ,,cos θ ) ad (si θ,si θ,,si θ ) ae liealy depedet Summig up the above cases, we fiish the poof of this theoem = Refeeces [] Hog-pig, M A, Zheg-ke, MIAO, ad Jiog-sheg, L I, 008, Geealized omal matix, Appl Math JChiese uiv, 3(), pp40-44 [] Ho, R A, ad Johso, C R, 985, Matix Aalysis, Cambidge: Cambidge Uivesity pess, pp79-48 [3] Beckebach, E F, ad Bellma, R, 983 Iequalities, Beli: Spige, PP6- [4] Kishamoothy, S, ad Subash, R, 0 O k-omal matices Iteatioal J of Math Sci & Egg Appls Vol5, oii, pp9-30