Congruence Diagonalization of Two. Hermite Matrices Simultaneously

Transcription

1 International Journal of Algebra, Vol 4, 2010, no 23, Congruence Diagonalization of Two Hermite Matrices Simultaneously Junqing Wang, Jiyong Lu and Yumin Feng Department of Mathematics Tianjin Polytechnic University Tianjin , China Abstract Matrix diagonalization is very important in many areasin this paper, we mainly study the problem of simultaneous congruence diagonalization of two Hermite matrices in and give some good results Mathematics Subject Classification: 47H09; 47H10 Keywords: Hermite matrices; congruence; simultaneous diagonalization 1 Introduction Matrix diagonalization plays a very important role in algebra Especially there is a wide range of application in the matrix theory,quadratic and linear transform et The problem of single matrix diagonalization has been discussed systematically in many articles,however, one often encounter the problem of simultaneous diagonalization of two matrices In this paper, we mainly study problems of simultaneous congruence diagonalization of two matrices, and give some theorems and proofs Let C be a complex field C n n denotes the set of all n-order matrices over C ForA C n n, A H is said to be the conjugate transpose of A

2 1120 Junqing Wang, Jiyong Lu and Yumin Feng 2 Preliminaries Definition 21 Let A C n n,ifa H = A,then A is called a Hermite matrix Definition 22 Let U C n n,if U H U = UU H = I,U is called a Unitary matrix Definition 23 Let A C n n is a Hermite matrix,if for any x C n and x 0,x H Ax > 0 is permanent establishment,then A is called positive definite matrix, denoted A>0; if for any x C n, x H Ax 0 is permanent establishment,then A is called non-negative definite matrix, denoted A 0If A B 0,we say A is greater than or equal to B,denoted A B Lemma 24 matrix Let A C n n is a Hermite matrix,then A 1 is also a Hermite Lemma 25 Let A C n n is a Hermite matrix,then S H AS is also a Hermite matrix for an arbitrary phalanx S Lemma 26 Let A C n n is a Hermite matrix,then A>0 if and only if there exists an invertible matrix P to make P H AP = I be satisfied Lemma 27 Let A C n n is a Hermite matrix,then there exists an Unitary matrix U to make U H AU be diagonal matrix 3 Conclusions Theorem 31 Let A, B C n n are two Hermite matrices,and B>0,then there exists an invertible matrix P to make P H AP and P H BP both diagonal matrices Proof For B > 0,according to the Lemma 26,there exists an invertible matrix P 1 to make P1 HBP 1 = I And A is a Hermite matrix,with Lemma 25,P1 H AP 1 is also a Hermite matrix,then there exists an Unitary matrix U to make U H (P1 HAP 1)U = diag(λ 1,λ 2,λ n ) be satisfied Put P = P 1 U,then P is invertible and P H BP = U H P1 H BP 1U = U H IU = I,P H AP = U H P1 H AP 1U = diag(λ 1,λ 2,λ n ) So the conclusion can be true Theorem 32 suppose A, B C n n are two Hermite matrices,then there exists an Unitary matrix U to make U H AU and U H BU both diagonal matrices if and only if AB = BA

3 Congruence diagonalization 1121 Proof Proof of the necessary conditions According to the known condition,we have U H AU =Λ 1,U H BU =Λ 2, where Λ 1 and Λ 2 are diagonal matrices with the eigenvalues of A and B as its diagonal elements respectively We can get A =(U 1 ) H Λ 1 U 1 and B = (U 1 ) H Λ 2 U 1,soAB =(U 1 ) H Λ 1 Λ 2 U 1 and BA = U 1 ) H Λ 2 Λ 1 U 1 Another part,for Λ 1 and Λ 2 are both diagonal matrices,λ 1 Λ 2 =Λ 2 Λ 1,therefore,we can obtain AB = BA Proof of sufficient conditions For A is a Hermite matrix,with Lemma 27,there exists an Unitary matrix U to make U H 0 λ AU = 0 0 λ s I ns be satisfied,where s i=1 n i = n andλ i (i =1, 2,,s)are different eigenvalues with each other of A and the multiplicity of λ i is n i Put U H BU,we prove U H BU to be a block diagonal matrix below B 11 B 12 B 1s suppose U H B BU = 21 B 22 B 2s For AB = BA and U H U = B s1 B s2 B ss UU H = I,so U H ABU = U H BAU,namely (U H AU)(U H BU) =(U H BU)(U H AU),that is, λ 1 B 11 λ 1 B 12 λ 1 B 1s λ 1 B 11 λ 2 B 12 λ s B 1s λ 2 B 21 λ 2 B 22 λ 2 B 2s λ = 1 B 21 λ 2 B 22 λ s B 2s λ s B s1 λ s B s2 λ s B ss λ 1 B s1 λ 2 B s2 λ s B ss Comparing the two matrices on both sides of the equation above,due to λ i λ j B (while i j),so B ij =0(i j)thus we have U H 0 B BU = 2 B 0 0 B ss is a Hermite matrix,then U H BU is also a Hermite matrix,so B ii (i =1, 2,s) are sub-block Hermite matricesfor each sub-block matrix B ii (i =1, 2,,s),there exists an Unitary matrix V i to satisy Vi H B ii V i =Λ i,where Λ i is a diagonal ma-

4 1122 Junqing Wang, Jiyong Lu and Yumin Feng V V trixtherefore we put V = and V is a Unitary matrix 0 0 V s apparently,so V 1 HB 11V Λ V H (U H 0 V H BU)V = 2 B 22 V 0 Λ = 0 0 Vs HB ssv s 0 0 Λ s and V H (U H AU)V = V 1 Hλ 1I n1 V V H 2 λ 2 I n2 V 0 λ = 0 0 Vs Hλ si ns V s 0 0 λ s I ns Note W = UV,then W is the Unitary matrix which can make W H AW,W H BW both diagonal matrices simultaneously Theorem 33suppose A, B C n n are two Hermite matrices, A, B 0,then there exists an invertible matrix P to make P H AP and P H BP both diagonal matrices Proof A, B C n n are two Hermite matrices,then A + B is also a Hermite matrix,therefore there exists an invertible matrix P 1 to make P1 H (A + B)P 1 = I r 0 be satisfied (where r = rank(a+b))put P1 H 0 0 BP 1 = B 11 B 12 B 21 B 22 (where B 11 is a r r phalanx),we have P1 H (A+B)P 1 0 and P1 HBP 1 0 and P1 H(A + B)P 1 P1 HBP 1,that is, I r 0 B 11 B 12, sob 12 = B 21 = 0 0 B 21 B 22 B 22 = 0 and B 11 0However B 11 is also Hermite matrix,therefore there exists an Unitary matrix U tomake U H B 11 U =Λ r (where Λ r is a r-order diagonal matrix)we put P = P 1 U 0,thenP H BP = Λ r 0 and P H AP = 0 I n r 0 0 P H (A+B B)P = P H (A+B)P P H BP = I r Λ r 0,therefore P is the 0 0 invertible matrix which can make P H AP and P H BP both diagonal matrices

5 Congruence diagonalization 1123 Theorem 34 suppose A, B C n n are two Hermite matrices,and A is invertible,then A and B can be congruence diagonalized simultaneously if and only if A 1 B can be diagonalized ProofProof of the necessary conditions According to the known condition, there exists an invertible matrix P which make A =(P 1 ) H Λ 1 P 1 and B =(P 1 ) H Λ 2 P 1 be satisfied (where Λ 1 and Λ 2 are both diagonal matrices)then A 1 B = P Λ 1 1 P H (P 1 ) H Λ 2 P 1 = P Λ 1 1 Λ 2 P 1 = P ΛP 1 (where Λ = Λ 1 1 Λ 2 is a diagonal matrix),therefore P 1 A 1 BP = Λ,the conclusion come true Proof of sufficient conditions A 1 B can be diagonalized,so there exists an invertible matrix P to make P 1 A 1 0 λ BP = =Λ λ s I ns be satisfied (where s i=1 n i = n andλ i (i = 1, 2,,s)are different eigenvalues with each other of A 1 B and the multiplicity of λ i is n i )Therefore BP = AP Λ 1,we make both sides of the equation BP = AP Λ 1 be multiplied by P H,then we can get P H BP = P H AP Λ 1 For P H BP is also a Hermite matrix,then P H BP = (P H BP) H,that is,p H AP Λ 1 = Λ H 1 P H A H P = Λ 1 P H AP,so with the proof of Theorem 32 we can know that P H AP is a block diagonal matrix Set P H AP =Λ 2,there exists an block Unitary matrix V V V = to make V H (P H AP )V =Λ 2 (where Λ 2 is a diagonal matrix)put Q = PV,then Q H AQ = V H P H AP V =Λ 2 and Q H BQ = 0 0 V s V H P H BPV = V H P H AP Λ 1 V = V H P H AP V Λ 1 =Λ 2 Λ 1 = Λ(where Λ is also a diagonal matrix)therefore Q is the invertible matrix which can make A and B be congruence diagonalization simultaneouslythe conclusion be established Theorem 35 suppose A, B C n n are two Hermite matrices,and A, B are non-invertible,however there exists a λ 0 C to make A + λ 0 B invertible,then A and B can be congruence diagonalized simultaneously if and only if (A + λ 0 B) 1 B can be diagonalized ProofProof of the necessary conditions

6 1124 Junqing Wang, Jiyong Lu and Yumin Feng According to the known condition, there exists an invertible matrix P,which make A =(P 1 ) H Λ 1 P 1 and B =(P 1 ) H Λ 2 P 1 be satisfied (where Λ 1 and Λ 2 both diagonal matrices)then (A+λ 0 B) 1 B = P (Λ 1 +λ 0 Λ 2 ) 1 P H (P 1 ) H Λ 2 P 1 = P (Λ 1 + λ 0 Λ 2 ) 1 Λ 2 P 1 = P ΛP 1 (where Λ = (Λ 1 + λ 0 Λ 2 ) 1 Λ 2 is a diagonal matrix),therefore P 1 (A + λ 0 B) 1 BP = Λ,the conclusion come true Proof of sufficient conditions(a+λ 0 B) 1 B can be diagonalized,so there exists an invertible matrix P to make P 1 (A+λ 0 B) 1 0 λ BP = = 0 0 λ s I ns Λ 1 be satisfied (where s i=1 n i = n and λ i (i =1, 2,,s)are different eigenvalues with each other of (A+λ 0 B) 1 B and the multiplicity of λ i is n i )Therefore BP =(A+λ 0 B)P Λ 1,we make both sides of the equation BP =(A+λ 0 B)P Λ 1 be multiplied by P H,then we can get P H BP = P H (A + λ 0 B)P Λ 1 For P H BP is also a Hermite matrix,then P H BP =(P H BP) H,that is,p H (A+λ 0 B)P Λ 1 = Λ H 1 P H (A + λ 0 B) H P =Λ 1 P H (A + λ 0 B)P,so we can know that P H (A + λ 0 B)P is a block diagonal matrix Set P H (A + λ 0 B)P =Λ b,there exists an block V V Unitary matrix V = to make V H [P H (A + λ 0 B)P ]V = 0 0 V s Λ 2 (where Λ 2 is a diagonal matrix)put Q = PV,then Q H BQ = V H P H BPV = V H P H (A + λ 0 B)P Λ 1 V = V H P H (A + λ 0 B)PVΛ 1 =Λ 2 Λ 1 = Λ(where Λ is also a diagonal matrix)and Q H AQ = Q H (A + λ 0 B λ 0 B)Q = Q H (A + λ 0 B)Q Q H λ 0 BQ =Λ 2 λ 0 Λ(where Λ 2 λ 0 Λ is also a diagonal matrix)thus Q is the invertible matrix which can make A and B be congruence diagonalization simultaneouslythe conclusion be established Theorem 36 Two 2 2 Hermite matrices whose ranks are both one must be congruence diagonalized simultaneously ProofLet A and B are the two matrices satisfying the known conditionaccording to the condition,we can get only two situations,that is, either A = λb or A+B is invertiblewe discuss the two points below While A = λb,for A and B is Hermite matrices,apparently there exists an invertible matrix P which make P H AP and P H BP both diagonalization

7 Congruence diagonalization 1125 simultaneously While A + B is invertible,we know A + B and(a + B) 1 and (A + B) 1 B are also Hermite matrices Especially for (A + B) 1 B,we can put invertible matrices P which make P H (A + B) 1 BP be a diagonalization matrix,therefore with the Theorem 35,we can get A and B can be congruence diagonalized simultaneously References [1] Qiu Wei-sheng,The suficient and necessary condition for simultaneously congruent diagonalization of two symmetric matrice[j],journal of LONGYAN Univercity, Vol 26, 2008 [2] Chen Xianping et,the conditions of simultaneous diagonalization of two matrices[j](in Chinese),Journal of ZAOZHUANG Univercity, Vol 122, No 2, 2005 [3] Zhou Liren, On the condition about simultaneous diagonalization of matrixes, Journal of Hunan Institute of Science and Technology, Vol 20, No 1,March 2007 [4] Xia Xuan, Diagonalization of two matrices simultaneously, Journal of Nan- Chang Institute of Aeronautical Technology (Natural Sciences), Vol 17, No 3, Sep 2003 [5] Wang Xinzhe and Jiang Yan-jie, The discussion on generalized diagonalization matrix[j], College Mathematics, Vol 25, August 2009 Received: June, 2010