International Journal of Computational Science and Mathematics. ISSN 0974-3189 Volume 3, Number 4 (2011), pp. 411-420 © International Research Publication House http://www.irphouse.com

Generalized k-normal Matrices

1S. Krishnamoorthy and 2R. Subash

1Department of Mathematics, Government Arts College (Autonomous), Kumbakonam, Tamilnadu, India 612 001. 2Department of Mathematics, A.V.C. College of Engineering, Mannampandal, Mayiladuthurai, Tamilnadu, India 609305. E-mail: [email protected]

Abstract

A matrix A ∈ ^ nn× is called generalized k-normal provided that there is a positive definite k-hermite matrix H such that HAH is k-normal. In this paper, these matrices are investigated and their canonical form, invariants and relative properties in the sense of congruence are obtained.

AMS classification: 15A04, 15A18.

Keywords: Generalized k-normal matrix, congruence, canonical form and invariant.

Introduction

Let ^ nn× be the space of nxn complex matrices. Let ‘k’ be a fixed product of disjoint transposition in Sn={1,2,…n} (hence, involutary) and let K be the associated

permutation matrix of k. Two matrices AB, ∈ ^ nn× are called k-unitarly similar provided that there exists a k-unitarly matrix U such that B = KU∗ AU , where U ∗ is

the conjugate transpose matrix of U. For a matrix A ∈ ^ nn× , there exists a k-unitary matrix U such that KU∗ AU is an upper triangular matrix and A is k-unitarly similar

to a k-diagonal matrix if and only if A is k-normal. Two matrices AB, ∈ ^ nn× are ∗ congruent if there exists a nonsingular matrix P ∈ ^ nn× such that B = KP AP. It is

easy to see that, for a given matrix A ∈ ^ nn× , there exists a nonsingular matrix 412 S. Krishnamoorthy and R. Subash

∗ P ∈ ^ nn× such that KP AP is an upper triangle matrix since k-unitary similar matrices A, B are congruent. In this paper, we obtain the canonical form and invariants for generalized k- normal matrices in the sense of congruence. We give some determinantal inequalities for generalized k-normal matrices and their k-hermite part and skew k-hermite part.

Basic Definitions Definition 2.1: A matrix A ^ is said to be k-hermitian, if AKAK= ∗ . ∈ nn× That is, i,j=1,2,…,n. aaij= k() j k () i ,

⎡ 01 0⎤ ⎢ ⎥ Example 2.2: Ai= ⎢ 00 ⎥ is k-hermitian. ⎣⎢−i 00⎦⎥

∗ Definition 2.3: A matrix A ∈ ^ nn× is said to be skew k-hermitian, if AKAK=− . That is, , i,j=1,2,…,n . aaij=− k() j k () i

⎡ 00− i ⎤ ⎢ ⎥ Example 2.4: Ai= ⎢ 00⎥ is skew k-hermitian. ⎣⎢ 00−i ⎦⎥

Generalized k-normal matrices Definition 3.1[4]: A matrix A ^ is said to be k-normal, if AA∗∗ K= KA A. ∈ nn×

That is, aaij n−+ k()1 j k () i= a k () j n −+ k ()1 i a ij; i, j = 1, 2… n.

⎡ 03−−ii⎤ Example 3.2: Aii=−⎢ −30⎥ is k-normal. ⎢ ⎥ ⎣⎢−−30ii⎦⎥

∗∗ Definition 3.3[4]: A matrix A ∈ ^ nn× is said to be k-unitary, if AA K== KA A K.

⎡ i 1 ⎤ ⎢ 22⎥ Example 3.4: A = ⎢ ⎥ is k-unitary. ⎢ 1 i ⎥ ⎢ ⎥ ⎣ 22⎦

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Definition 3.5: A matrix A ∈ ^ nn× is called generalized k-normal if there is a positive definite k-hermite matrix H such that HAH is k-normal.

⎡⎤11− i Example 3.6: H = ⎢⎥is a positive definite k-hermite matrix. ⎣⎦11+ i ⎡⎤−i 1 Then A = ⎢⎥ is generalized k-normal. ⎣⎦0 i

Theorem 3.7: A ∈ ^ nn× is generalized k-normal if and only if there exists a positive definite k-hermite matrix X such that KAXA∗∗= AXAK.

Proof: Assume that A is generalized k-normal matrix. Then by definition (3.5), there exists a positive definite k-hermite matrix H such that HAH is k-normal. Therefore, 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∗∗() KH22 AH HA () HK AKH ⇒=KA∗∗() KH22 A A () HK AK (1)

Since H is a positive definite, H is non-singular. Then H2 is positive definite. Let XH==222 HKHK =()() 2 = KH 2.Then X is a positive definite k-hermite matrix. There fore (1) ⇒ KAXA∗∗= AXAK. Conversely, Assume that KAXA∗∗= AXAK and X is a positive definite k- hermite matrix. Then there exists a positive definite k-hermite matrix H such that XH==222 HKHK =()() 2 = KH 2. Thus ∗∗22 KA() KH A= A () HK AK ∗∗ ⇒=HKAKHKHAH HAHKHKAKH ∗∗∗ ∗∗∗ ⇒=()KHKKAHHAH HAHHAKKHK () ∗∗ ∗ ∗∗ ∗ ⇒=KH AH HAH HAHH AH K ∗∗ ⇒=KHAH()()()() HAH HAHHAHK Hence HAH is k-normal and then A is generalized k-normal.

Theorem 3.8: A ∈ ^ nn× is generalized k-normal if and only if A is congruent to a k- diagonal matrix.

414 S. Krishnamoorthy and R. Subash

Proof: Assume that A is generalized k-normal matrix. Then by definition (3.5), there exists a positive definite k-hermite matrix H such that HAH is k-normal. By Schur’s theorem [2], there exists a k-unitary matrix U such that KU∗ () H AH U is k-diagonal matrix. Let P = HU ⇒ P∗∗∗∗==UH UKHK ⇒ P∗∗H= U KHKH ⇒ P∗∗HUKH= ()2 ⇒ P∗∗HUHK= 22 ⇒ P∗∗HUH= 2 ⇒ P∗∗= UH. Then P is nonsingular and KP∗ AP is k-diagonal matrix. That is, A is congruent to a k-diagonal matrix. Conversely, Assume that A is congruent to a k-diagonal matrix. Then there exists a nonsingular matrix P such that KP∗ AP= Dis a k-diagonal matrix. By the polar factorization theorem of matrices, there exists a positive definite k-hermite matrix H and a k-unitary matrix U such that P = HU ⇒ P∗∗∗∗==UH UKHK ⇒ P∗∗H= U KHKH ⇒ P∗∗HUKH=⇒()2 P∗∗HUHK=⇒22 P∗∗HUH=⇒2 P∗∗= UH. Thus KU∗ () H AH U= Dis a k-diagonal matrix. By Schur’s theorem [2] and definition (3.5), A is generalized k-normal.

Theorem 3.9: Let A ∈ ^ nn× be a generalized k-normal matrix. Then there exists a

∗ ⎛⎞D 0 iiθθ iθ nonsingular matrix P such that A = KP⎜⎟ P, where Ddiagee= (12 , ,..., er ) ⎝⎠00 iiθθ12 iθ r , r = rank(A), −≤π θθ12 ≤ ≤.... ≤ θπθθπrr < , − 1 < and ee, ,..., e are all nonzero k-eigen values of P−∗−11()KP A.

Proof: Since A is generalized k-normal matrix. By Theorem (3.8) there exists a ∗ nonsingular matrix Q such that AKQdiag= (λλ12 , ,..., λr ,0,0,...,0) Q, where r = rank(A) is the rank of A, λ12,λλ ,..., r are nonzero complex numbers. ejriθ j ,0,1,2,..., Let λρjj=>= ρ j and −π ≤≤≤≤<θθ12.... θπr . 11 1 22 2 ∗ ∗∗ Denote Rdiag= (ρρ12 , ,..., ρr ,1,...,1) and P =⇒=RQ P Q R .

∗ ⎛⎞D 0 Then P is nonsingular and A = KP⎜⎟ P. ⎝⎠00

−∗−−∗−∗1111⎛⎞DDD000 ⎛⎞ ⎛⎞ Since P ⎜⎟()KP A= P ⎜⎟ ()() KP K P ⎜⎟ P ⎝⎠00 ⎝⎠ 00 ⎝⎠ 00 2 −1 ⎛⎞D 0 22iiθθ 2iθ = P ⎜⎟P, ee12, ,..., er are all nonzero k-eigen values of ⎝⎠00

−∗−11⎛⎞D 0 P ⎜⎟()KP A. ⎝⎠00

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Without loss of generality, −≤π 2θθ ≤ 2 ≤ .... ≤ 2 θπ < (mod 2 π ). jj12 jr Then 222θ −≤θθ − 2 θ < 2(mod2). π πThus θ −θπ< . rjrj11 r 1

∗ ⎛⎞D 0 Since A = KP⎜⎟ P, ⎝⎠00

−∗−−∗−∗1111⎛⎞DD00 − 1 ⎛⎞ P ()KP A== P ()() KP K P⎜⎟ P P ⎜⎟ P . ⎝⎠00 ⎝⎠ 00 Hence eeiiθθ12, ,..., eiθ r are all nonzero k-eigen values of P−∗−11()KP A.

Theorem 3.10: Let A ∈ ^ nn× be a generalized k-normal matrix. If there exist a  ∗∗⎛⎞D 0 ⎛⎞D 0 nonsingular matrix P and Q such that AKP==⎜⎟ PKQ⎜⎟ Q, where ⎝⎠00 ⎝⎠00 Ddiagee(iiθθ12 , ,..., eiθ r ), = −≤π θθ12 ≤ ≤.... ≤ θπθθπrr < , − 1 < , r = rank(A),  iw12 iw iwr Ddiagee= ( , ,..., e ), −≤π ww12 ≤ ≤.... ≤ wr <π and wwr − 1 < π . Then for 1.j r θ jj= w ≤≤

 ∗∗⎛⎞D 0 ⎛⎞D 0 Proof: Given AKP==⎜⎟ PKQ⎜⎟ Q ⎝⎠00 ⎝⎠00  ∗∗⎛⎞D 0 ⎛⎞D 0 ⇒=K PPKQQ⎜⎟ ⎜⎟. ⎝⎠00 ⎝⎠00

Pre multiply by K on both sides.  ∗∗⎛⎞D 0 ⎛⎞D 0 Therefore, P ⎜⎟PQ= ⎜⎟ Q ⎝⎠00 ⎝⎠00  ⎛⎞D 0 ∗−11 ∗⎛⎞D 0 − ⇒=⎜⎟()P QQP⎜⎟ (2) ⎝⎠00 ⎝⎠00

⎛⎞RR11 12 DenoteQP−1 == R , where R is an rxr complex matrix. ⎜⎟11 ⎝⎠RR21 22 ∗∗ ⎛⎞RR11 12 ⇒===()()QP−∗11 P ∗− Q ∗ R ∗ ⎜⎟ ⇒ ⎜⎟∗∗ ⎝⎠RR21 22 ∗∗ ⎛⎞D 0 ⎛⎞RR11 12⎛⎞D 0 ⎛⎞ RR 11 12 ⎛⎞D 0 (2) ⇒=⎜⎟⎜⎟⎜⎟⎜⎟⇒ ⎜⎟ 00 ⎜⎟∗∗ ⎜⎟RR 00 ⎝⎠RR21 22 ⎝⎠00 21 22 ⎝⎠ ⎝⎠⎝⎠ ∗  ∗∗ ⎛⎞RD110 ⎛⎞ R 11 R 12 ⎛⎞RDR11 11 RDR 11 12 = ⎜⎟⎜⎟= ⎜⎟. ⎜⎟∗  ⎜⎟RR ⎜⎟∗∗ ⎝⎠RD21 0 ⎝⎠21 22 ⎝⎠RDR21 11 RDR 21 12

416 S. Krishnamoorthy and R. Subash

∗  ∗ Thus D = RDR11 11 . Since D is nonsingular, R11 and R11 are also nonsingular. Hence 21∗− ∗ − 111 − ∗ − ∗ ∗  ∗− 1121 ∗ − ∗  ∗ − DDD==() ( RDRRDR11 11 )( 11 ()) 11 = RDDR 11 ()() 11 = RDR 11 ( 11 ). That is, D2 and D 2 are similar. Thus D2 and D 2 have the same k-eigen values. 22iiw That is, eeθ jj= for1 ≤≤j r .

Therefore, 22(mod2)θ jj≡ w π for 1 ≤ j ≤ r and henceθ jj≡ w (modπ ) for

1 ≤≤j r . By the conditions of the theorem, θ jj= w for1 ≤ j ≤ r .

Remark 3.11: Theorem (3.10) indicates that when a generalized k-normal matrix A

∗ ⎛⎞D 0 iiθθ12 iθ r satisfies A = KP⎜⎟ P, then Ddiagee= ( , ,..., e ) is independent of the ⎝⎠00 choice of the nonsingular matrix P, that is, the k-diagonal matrix D is only determined by A. So we have the following definitions.

Definition 3.12: Let A be a generalized k-normal matrix satisfying

∗ ⎛⎞D 0 iiθθ iθ A = KP⎜⎟ P.Denote DKA( )== D diage (12 , e ,..., e r ) , where r = ⎝⎠00 ⎛⎞D 0 rank(KA), −≤π θθ12 ≤ ≤.... ≤ θπr

Theorem 3.13: Let AB, ∈ ^ nn× be generalized k-normal matrices. Then A and B are congruent if and only if σ kk()AB= σ (), that is A and B have the same generalized k- spectrum.

Proof: Assume that A and B are congruent. Then there exists a nonsingular matrix Q such that B = KQ∗ AQ. By Theorem (3.9), there exists a nonsingular matrix P such that

∗ ⎛⎞DA() 0 A = KP⎜⎟ P. ⎝⎠00

∗∗⎛⎞DA() 0 Thus B = KQ K P⎜⎟ PQ. (3) ⎝⎠00

Denote R = PQ,then R is nonsingular. ⇒ R = PKQK ⇒ R∗ = KQ∗∗ KP .

Generalized k-normal Matrices 417

∗ ⎛⎞DA() 0 ∗ ⎛⎞DKA()0 Therefore (3) ⇒ B = RR⎜⎟ ⇒=KB R⎜⎟ R . ⎝⎠00 ⎝⎠00 By a Theorem (3.10), D(KB) = D(KA). Hence σ ()KAKB= σ () ()AB () ()AB () ⇒=σ kkσ . Conversely, Let σ kk= σ ⇒=σ ()KAKBσ () ⇒ D(KA) = D(KB) = D and there exists a nonsingular matrices P , Q such that

∗ ⎛⎞D 0 A = KP⎜⎟ P (4) ⎝⎠00

∗ ⎛⎞D 0 and B = KQ⎜⎟ Q (5) ⎝⎠00

Pre multiply by ()KP∗−1 and post multiply by P−1 in (4). Therefore (4) ⇒

∗−11 − ⎛⎞D 0 ∗∗−−11 ()KP AP = ⎜⎟ .Hence B = ()()KQ KP AP Q. Therefore A and B are ⎝⎠00 congruent.

Determinantal properties 1 Let A ∈ ^ . Alternatively, let H ()AAKAK=+ (∗ ), the k-hermite part and nn× 2 1 SA()=− ( A KAK∗ ),the skew k-hermite parte. It is known that AHASA=+() () 2 and KAK∗ =− H() A S () A . A is called positive definite provided that H(A) is positive definite k-hermite matrix. It is easy to know that A is positive definite if and ∗ T only if Re(Kx , Ax )> 0 for each nonzero complex column vector x = (xx12 , ,..., xn ) , where Re(a) denotes the real part of a complex number ‘ a ’.

Theorem 4.1: Let A ∈ ^ nn× be a positive definite matrix, then A is generalized k- normal.

Proof: since A is positive definite, H(A) is nonsingular. Then KAKHA∗−[()]11 A=− [() HA SA ()][()][() HA − HA + SA ()]

−−11 =−{HA ( )[ HA ( )] SA ( )[ HA ( )] }[ HA ( ) + SA ( )] =−{()[()]}[()()]I SA HA−1 HA + SA =+−HA() SA () SA ()[()][() HA−1 HA + SA ()]

−−11 =+−HA() SA () SA ()[()]() HA HA − SA ()[()]() HA SA

418 S. Krishnamoorthy and R. Subash

−1 =HA() +−− SA () SA () SA ()[()]() HA SA =−HA() SA ()[()]() HA−1 SA (5) −∗11 − AHA[ ( )] KAK=+ [ HA ( ) SA ( )][ HA ( )] [ HA ( ) − SA ( )] −1 =+{I SA ( )[ HA ( )] }[ HA ( ) − SA ( )] −1 =−+HA() SA () SA ()[()][() HA HA − SA ()] −1 =−+−HA() SA () SA () SA ()[()]() HA SA =−HA() SA ()[()]() HA−1 SA (6)

From (5) and (6), we get KA∗− K[()] H A1 A = AH[()] A−∗1 KAK (7)

Since H is a positive definite k-hermite matrix. Hence [()]HA −1 is a positive definite k-hermite matrix. By a Theorem (3.7), X== KHA[()][()]−−11 HA Kis a positive definite k-hermite matrix. Therefore (7)⇒ KA∗ XA = AXA∗ K . Hence A is generalized k-normal matrix.

Theorem 4.2: Let ‘n’ be an integer with n ≥ 3 and A ∈ ^ nn× a generalized k-normal matrix. Then det (AHASA )≥+ det ( ) det ( ) . Further, the equality holds if and only if r = rank(A)

∗ ⎛⎞D 0 Proof: There exists a nonsingular matrix P such that A = KP⎜⎟ P . It is easy ⎝⎠00 to see that HA()= KPHDP∗ () and SA()= KPSDP∗ (). Thus rank [H(A)]=rank [H(D)] and rank [S(A)]= rank[S(D)]. We distinguish the following cases.

Case (1): If rank (A)

Case (2): If rank (A)=n. Now, Ddiagee= (iiθθ12 , ,..., eiθ n ),

⇒=Ddiag(cosθ 1122 + i sinθθ ,cos + i sin θ ,...,cos θnn + i sin θ ) .

⇒=H( D ) diag (cosθ 12 ,cosθθ ,...,cosn ) , SD( )= idiag (sinθ 12 ,sinθθ ,...,sinn ) and n n ∗ 2 iθ j 2 AKPDP= .Then det(AKPe )=Π (det ) . ⇒=ΠdetHA ( ) (det KP ) cosθ j and j=1 j=1 n n 2 detSA ( )=Π i (det KP ) sinθ j . j=1

Generalized k-normal Matrices 419

Sub case (2.1): If cosθ j = 0 for some j. n 22 Then detH (ASAKP )+=Π≤≤ det ( ) det sinθ j det KPA det ( ) . Further, the j=1 above equality holds if and only if sinθ j = 1for all j, if and only if cosθ j = 0 for all j, if and only if H(A)=0 if and only if A is skew k-hermitian.

Sub case (2.2): If sinθ j = 0for some j. n 22 Then detH (ASAKP )+=Π≤≤ det ( ) det cosθ j det KPA det ( ) . Further, the j=1 above equality holds if and only if cosθ j = 1for all j, if and only if sinθ j = 0for all j, if and only if S(A)=0 if and only if A is k-hermitian.

Sub case (2.3): If cosθ j ≠ 0 and sinθ j ≠ 0for all j. nn 2 ⎛⎞ Now detHA ( )+= det SA ( ) det KP ⎜⎟ Π+Π cosθ jj sinθ ⎝⎠jj==11 22 ⇒+=detHA ( ) det SA ( ) det KP ( cosθθ12 cos + sin θθ 12 sin ) ≤= det KP det ( A ) By summing up the above cases the proof of this theorem is completed.

Remark 4.3: Let x12,xxyyy ,...,nn , 12 , ,... be non negative real numbers. Then 11111 nnnnn (x12 ,xx ,...,nn )+≤+++ ( yyy 12 , ,... ) ( xyxyxy 1 1 ) ( 2 2 ) ...( nn ) and the equality holds if and only if there exists and integer j such that xyjj= = 0 or (x12 ,xx ,...,n ) and

(yy12 , ,... yn ) are linearly dependent.

Theorem 4.4: Let ‘n’ be an integer with n ≥ 3 and A ∈ ^ nn× a generalized k-normal

iiθθ12 iθ r matrix with generalized k-spectrum σ k (Aee )= ( , ,..., e ), where r = rank(A). 222 Then det (AHASA )nnn≥+ det ( ) det ( ) .Further, the equality holds if and only if 22 2 22 2 r

∗ ⎛⎞D 0 Proof: There exists a nonsingular matrix P such that A = KP⎜⎟ P.It is easy to ⎝⎠00 see that HA()= KPHDP∗ () and SA()= KPSDP∗ (). Thus rank [H(A)]=rank [H(D)] and rank [S(A)]= rank[S(D)]. We distinguish the following cases.

Case (1): If rank (A)

420 S. Krishnamoorthy and R. Subash

(A)

⇒=H( D ) diag (cosθ 12 ,cosθθ ,...,cosn ) , SD( )= idiag (sinθ 12 ,sinθθ ,...,sinn ) and n n ∗ 2 iθ j 2 AKPDP= .Then det (AKPe )=Π (det ) . ⇒=ΠdetHA ( ) (det KP ) cosθ j and j=1 j=1 n n 2 detSA ( )=Π i (det KP ) sinθ j . j=1 nn 2 ⎛⎞ Thus detHA ( )+= det SA ( ) det KP ⎜⎟ Π+Π cosθ jj sinθ ⎝⎠jj==11 224nn11 ⎛⎞22 nnn nn ⇒+=Π+ΠdetHA ( ) det SA ( ) det KP ⎜⎟ (cosθθjj ) (sin ) . ⎝⎠jj==11

222 By Remark (4.3), detH (ASAA )nnn+≤ det ( ) det ( ) and the equality holds if 22 and only if there exists an integer j such that cosθθjj== sin 0 or 22 2 22 2 (cosθ12 ,cosθθ ,...,cosn ) and (sinθ12 ,sinθθ ,...,sinn ) are linearly dependent. 22 22 2 Since cosθθjj+= sin 1, the equality holds if and only if (cosθ12 ,cosθθ ,...,cosn ) 22 2 and (sinθ12 ,sinθθ ,...,sinn ) are linearly dependent. Summing up the above cases, we finish the proof of this theorem.

References

[1] Hong-ping, M. A., Zheng-ke, MIAO, and Jiong-sheng, L. I., 2008, “Generalized normal matrix”, Appl. Math. J.Chinese univ., 23(2), pp.240-244. [2] Horn, R. A., and Johnson, C. R., 1985, Matrix Analysis, Cambridge: Cambridge University press, pp.79-482. [3] Beckenbach, E. F., and Bellman, R., 1983 “ Inequalities,” Berlin: Springer, PP.26- [4] Krishnamoorthy, S., and Subash, R., 2011 “On k-normal matrices” International J. of Math. Sci. & Engg. Appls. Vol.5, no.II , pp.119-130.