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volume 7, issue 3, article 116, 2006.

Received 01 March, 2006;

accepted 20 June, 2006.

Communicated by:B. Yang

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Journal of Inequalities in Pure and Applied Mathematics

SOME INEQUALITIES FOR SPECTRAL VARIATIONS

SHILIN ZHAN

Department of Mathematics Hanshan Teacher’s College Chaozhou, Guangdong China, 521041

EMail:shilinzhan@163.com

c

2000Victoria University ISSN (electronic): 1443-5756 060-06

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Some Inequalities for Spectral Variations

Shilin Zhan

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J. Ineq. Pure and Appl. Math. 7(3) Art. 116, 2006

http://jipam.vu.edu.au

Abstract

Over the last couple of decades, significant progress for the spectral variation of a matrix has been made in partially extending the classical Weyl and Lidskii theory [11,7] to normal matrices and even to diagonalizable matrices for example. Recently these theories have been established for relative perturbations.

In this paper, we shall establish relative perturbation theorems for generalized normal matrix. Some well-known perturbation theorems for normal matrix are extended. As applying, some perturbation theorems for positive definite matrix (possibly non-Hermitian) are established.

2000 Mathematics Subject Classification:15A18, 15A42, 65F15.

Key words: Spectral variation; Unitarily invariant norm; Hadamard product; Relative perturbation theorem.

1. Introduction

The set of allλ∈ Cthat are eigenvalues ofA ∈M_n(C)is called the spectrum of Aand is denoted by σ(A). The spectral radius ofAis the nonnegative real number ρ(A) = max{|λ| : λ ∈ σ(A)}. We shall usek|·|kto denote a unitarily invariant norm (see [5, 9, 13, 3, 20, 21]). kXk₂, the largest singular value of X, is a frequently used unitarily invariant norm. LetX ◦Y = (x_ijy_ij)be the Hadamard product of X = (x_ij)andY = (y_ij). A matrixA ∈ M_n(C)is said to be a generalized normal matrix with respect to H(It is called “generalized normal matrix” for short) or H⁺-normal if there exists a positive definite Hermitian matrix H such that A^∗HA = AHA^∗, where “*” denotes the con- jugate transpose. The definition was given first by [19, 18]. A generalized normal matrix is a very important kind of matrix which contains two subclasses of important matrices: normal matrices and positive definite matrices (possibly non-Hermitian), where a matrixAis called normal ifA^∗A=AA^∗ and positive definite ifRe(x^∗Ax)>0for any non-zerox∈Cⁿ(see [5,6]). In recent years, the geometric significance, sixty-two equivalent conditions and many properties have been established for generalized normal matrices in [19,17,18]. We have Lemma 1.1 (see [19]). SupposeA∈M_n(C). Then

1. Ais a generalized normal matrix with respect toHif and only ifH^1/2AH^1/2 is normal.

2. A is a generalized normal matrix with respect to H if and only if there exists a nonsingular matrixP such thatH = (P P^∗)⁻¹ and

(1.1) A=PΛP^∗,

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whereΛ = diag(λ₁, λ₂, . . . , λ_n). Furthermore,λ₁, λ₂, . . . , λ_nareneigen- values ofHA.

Remark 1. (1.1) is equivalent toHA = P^−∗ΛP^∗ withP^−∗ = (P⁻¹)^∗, so we say that A has generalized eigen-decomposition (1.1), and λ₁, λ₂, . . . , λ_n are the generalized eigenvalues of matrixA.

The spectral variation of a matrix has recently been a very active research subject in both matrix theory and numerical linear algebra. Over the last couple of decades significant progress has been made in partially extending the classical Weyl and Lidskii theory [11,16] to normal matrices and even to diagonalizable matrices for example. This note will show how certain perturbation problems can be reformulated as simple matrix optimization problems involving Hadamard products. When A and A˜ are normal, we have shown one of many perturbation theorems that can be interpreted as bounding the norms of Q◦Z whereQis unitary andZ is a special matrix defined by the eigenalues (see [10]). In this paper, we shall extend the above result, and shall show how certain perturbation problems can be reformulated as generalized normal matrix optimization problems involving Hadamard products. Also, we study how generalized eigenvalues of a generalized normal matrixAchange when it is perturbed toA˜=D^∗AD, whereDis a nonsingular matrix. As applications, some perturbation theorems for positive definite matrices (possibly non-Hermitian) are established.

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2. Main Result

Suppose that Aand A˜are generalized normal matrices with respect to a com- mon positive definite matrixH, and have generalized eigen-decompositions (2.1) A=PΛP^∗ and A˜= ˜PΛ ˜˜P^∗,

where

(2.2) Λ = diag(λ₁, λ₂, . . . , λ_n) and Λ = diag(˜˜ λ₁,˜λ₂, . . . ,λ˜_n) and λ_i are the generalized eigenvalues ofA, andλ˜_i are the generalized eigenvalues ofA˜(i= 1,2, . . . , n).

NoticeH = (P P^∗)⁻¹ andH = ( ˜PP˜^∗)⁻¹,so(P⁻¹P˜)^∗(P⁻¹P˜) = ˜P^∗HP˜ = I, thenQ=P⁻¹P˜is unitary and

(2.3) P˜ =P Q

Define

(2.4) Z₁ =

λ_i−λ˜_jn i,j=1

. We have the following result.

Theorem 2.1. Suppose A and A˜ are H⁺-normal with generalized eigen- decomposition (2.1), Then

(2.5) ρ(H)⁻¹k|Q◦Z₁|k ≤ A−A˜

≤ρ(H⁻¹)k|Q◦Z₁|k, whereQ=P⁻¹P˜ is unitary andZ1is defined in Eq.(2.4).

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Proof. ForAandA˜having generalized eigen-decomposition (2.1), noticing that P˜ =P Q, whereQ=P⁻¹P˜is unitary,k|W Y|k ≤ kWk₂k|Y|kandk|Y Z|k ≤ k|Y|k kZk₂(see [9, p. 961]), we have

PΛP^∗−P QΛQ˜ ^∗P^∗

≤ kPk₂

Λ−QΛQ˜ ^∗

kP^∗k₂,

then

A−A˜

≤

H⁻¹ 2

Λ−QΛQ˜ ^∗ . Since

Λ−QΛQ˜ ^∗ =

ΛQ−QΛ˜

=k|Q◦Z1|k andkH⁻¹k₂ =ρ(H⁻¹),

(2.6)

A−A˜

≤ρ(H⁻¹)k|Q◦Z₁|k. On the other hand, we have

P⁻¹ 2

PΛP^∗−P QΛQ˜ ^∗P^∗

P^−∗

2 ≥

Λ−QΛQ˜ ^∗

=k|Q◦Z₁|k. Similarly forH = (P P^∗)⁻¹andkP⁻¹k₂ =kP^−∗k₂ =p

ρ(H), we obtain ρ(H)

A−A˜

≥ k|Q◦Z₁|k, hence

(2.7)

A−A˜

≥ρ(H)⁻¹k|Q◦Z₁|k.

The inequality (2.5) completes the proof by inequalities (2.6) and (2.7).

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In particular, if H = I is the identity matrix, then H⁺-normal matrices A andA˜are normal matrices, henceAandA˜have eigen-decomposition

(2.8) A =UΛU^∗ and A˜= ˜UΛ ˜˜U^∗, whereU andU˜ are unitary, and

Λ = diag(λ₁, λ₂, . . . , λ_n), Λ = diag(˜˜ λ₁,˜λ₂, . . . ,λ˜_n).

By Theorem2.1, we have

Corollary 2.2 (see [10]). IfAandA˜are normal matrices, then (2.9)

A−A˜

=k|Q◦Z₁|k, whereQ=U^∗U˜ andZ₁ =

λ_i−λ˜_jn i,j=1.

We denote the Cartesian decompositionX =H(X)+K(X), whereH(X) =

1

2(X +X^∗), andK(X) = ¹₂(X −X^∗). Letσ(H(A)) = {h₁, h₂, . . . , h_n}be ordered so thath₁ ≥h₂ ≥ · · · ≥h_n. Then we have some perturbation theorems for positie definite matrices which are discussed as follows.

Corollary 2.3. If A = H(A) +K(A) and A˜ = H( ˜A) +K( ˜A) are positive definite with generalized eigen-decomposition (2.1), andQ=P⁻¹P˜ is unitary, then

(2.10) h_nk|Q◦Z₁|k ≤ A−A˜

≤h₁k|Q◦Z₁|k, whereZ1 is defined in Eq.(2.4).

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Proof. SinceQ=P⁻¹P˜is unitary,H(A) =H( ˜A). It is easy to see that A^∗H(A)⁻¹A=AH(A)⁻¹A^∗

and

A˜^∗H( Ã)⁻¹A˜= ÃH( Ã)⁻¹A˜^∗.

So A and A˜ are generalized normal matrices with respect to H(A)⁻¹. It is easy to see that ρ(H(A)⁻¹)⁻¹ = h_n, ρ(H(A)) = h₁. Applying Theorem 2.1, inequality (2.10) completes the proof.

Let B, C ∈ M_n(C). Then [B, C] = BC −CB is called a commutator and[B, C]_H = BHC−CHB is called a commutator with respect toH. The matricesBandCare said to commute with respect toHiff[B, C]H = 0. kXk_F is the Frobenius norm.

Corollary 2.4. LetAandA˜beH⁺-normal matrices. IfAandA˜commute with respect toH, then

(2.11) ρ(H)⁻¹k|I◦Z₁|k ≤ A−A˜

≤ρ(H⁻¹)k|I◦Z₁|k, whereI is the identity matrix, andZ₁ is defined in Eq.(2.4).

Proof. [A,A]˜_H = 0if and only if there exists a nonsingular matrixP, such that A =PΛP^∗ andA˜= PΛP˜ ^∗, whereQ =P⁻¹P =I (see [17, Theorem 3] and Theorem2.1). SoQis taken as the identity matrixIin Theorem2.1, hence Eq.

(2.11) holds.

Applying Corollary2.3and Corollary2.4, we have

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Corollary 2.5. Let the hypotheses of Corollary2.3hold. Moreover if matrices AandA˜commute with respect toH(A)⁻¹, then

(2.12) h_nk|I◦Z₁|k ≤ A−A˜

≤h₁k|I◦Z₁|k,

where h1 = max1≤i≤nλi(H(A)), hn = min1≤i≤nλi(H(A))andZ1 is defined in Eq.(2.4).

In the following, we shall study how generalized eigenvalues of a generalized normal matrix A change when it is perturbed to A˜ = D^∗AD, where D is a nonsingular matrix. Thep−relative distance betweenα,α˜ ∈Cis defined as (2.13) %p(α,α) =˜ |α−α|˜

pp

|α|^p+|α|˜ ^p for1≤p≤ ∞.

Theorem 2.6. Suppose A and A˜ are H⁺-normal matrices and A˜ = D^∗AD, where D is nonsingular. LetA and A˜ have generalized eigen-decomposition (2.1). Then there is a permutationτ of{1,2, . . . , n}such that

(2.14)

n

X

i=1

[%₂(λ_i,λ˜_τ(i))]² ≤c(kI−Dk²_F +

D^−∗−I

2 F)

wherec= max1≤i≤nλ_i(H)/min1≤i≤nλ_i(H).

Proof. Notice that

A−A˜=A−D^∗AD=A(I−D) + (D^−∗−I) ˜A.

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Pre- and postmultiply the equations byP⁻¹andP˜^−∗ respectively, to get (2.15) ΛP^∗P˜^−∗−P⁻¹P˜Λ = ΛP˜ ^∗(I−D) ˜P^−∗+P⁻¹(D^−∗−I) ˜PΛ.˜ SetQ=P⁻¹P˜ = (qij), thenQis unitary andQ=P^∗P˜^−∗. Let

(2.16) E =P^∗(I−D) ˜P^−∗ = (e_ij),E˜ =P⁻¹(D^−∗−I) ˜P = (˜e_ij).

Then (2.15) implies that ΛQ− QΛ = ΛE˜ + ˜EΛ˜ or componentwise λ_iq_ij − q_ijλ˜_j =λ_ie_ij + ˜e_ij˜λ_j, so

(λ_i−λ˜_j)q_ij

2

=

λ_ie_ij+ ˜e_ijλ˜_j

2 ≤(|λ_i|²+

˜λ_j

2)(|e_ij|²+|˜e_ij|²),

which yields[%₂(λ_i,λ˜_j)]²|q_ij|² ≤ |e_ij|²+|˜e_ij|².Hence

n

X

i,j=1

[%₂(λ_i,λ˜_j)]²|q_ij|²

≤

P^∗(I−D) ˜P^−∗

2 F

+

P⁻¹(D^−∗−I) ˜P

2 F

≤ kP^∗k²₂kI−Dk²_F

P˜^−∗

2 2

+ P⁻¹

2 2

D^−∗−I

2 F

P˜

2 2

.

Notice that

kP^∗k²₂ = max

1≤i≤nλ_i(H) and P⁻¹

2 2 =

1≤i≤nmin λ_i(H) −1

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byσ(P P^∗) = σ(P^∗P) = σ(H). Similarly, we have P˜

2

2 = max1≤i≤nλi(H) and

P˜^−∗

2 2

= max1≤i≤nλ_i(H⁻¹) =

1≤i≤nmin λ_i(H) ⁻¹

,

so n

X

i,j=1

h

%₂

λ_i,λ˜_ji2

|q_ij|² ≤c

kI−Dk²_F +

D^−∗−I

2 F

, wherec= max1≤i≤nλi(H)/min1≤i≤nλi(H).

The matrix |qij|²

n×n is a doubly stochastic matrix. The above inequality and [9, Lemma 5.1] imply inequality (2.14).

IfA andA˜are normal matrices, then they are generalized normal matrices with respect toH andH =I. Applying Theorem2.6, it is easy to get

Corollary 2.7. If A,A˜ ∈ M_n(C) are normal matrices withA = UΛU^∗ and A˜ = ˜UΛ ˜˜U^∗ where both U and U˜ are unitary, and A˜ = D^∗AD, where D is nonsingular, then

(2.17)

n

X

i=1

[%₂(λ_i,λ˜_τ(i))]² ≤ kI−Dk²_F +

D^−∗−I

2 F.

Corollary 2.8. Let A = H(A) + K(A)and A˜ = H( ˜A) +K( ˜A) be positive definite matrices with generalized eigen-decomposition (2.1), andA˜=D^∗AD, whereDis nonsingular. IfQ=P⁻¹P˜ is unitary, then

(2.18)

n

X

i,=1

h

%₂

λ_i,˜λ_τ(i)i2

≤c

kI−Dk²_F +

D^−∗−I

2 F

,

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wherec= max1≤i≤nλ_i(H(A))/min1≤i≤nλ_i(H(A)).

Proof. By the proof of Corollary2.3,AandA˜are generalized normal matrices with respect toH(A)⁻¹, and

1≤i≤nmaxλ_i(H(A)⁻¹)/ min

1≤i≤nλ_i(H(A)⁻¹) = max

1≤i≤nλ_i(H(A))/min1≤i≤nλ_i(H(A)).

Inequality (2.18) is proved by Theorem2.6.

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[9] REN-CANG LI, Relative perturbation theory: (1) eigenvalue and singular value variations, Technical Report UCB// CSD-94-855, Computer Sci- ence Division, Department of EECS, University of California at Berkeley, 1994. Also LAPACK working notes 85 (revised January 1996, available at http://www..netlib.org/lapack/lawns/lawn84.ps).

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