Perturbations of functions of diagonalizable matrices

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1 Electronic Journal of Linear Algebra Volume 20 Volume 20 (2010) Article Perturbations of functions of diagonalizable matrices Michael I. Gil Follow this and additional works at: Recommended Citation Gil, Michael I.. (2010), "Perturbations of functions of diagonalizable matrices", Electronic Journal of Linear Algebra, Volume 20. DOI: This Article is brought to you for free and open access by Wyoming Scholars Repository. It has been accepted for inclusion in Electronic Journal of Linear Algebra by an authorized editor of Wyoming Scholars Repository. For more information, please contact

2 PERTURBATIONS OF FUNCTIONS OF DIAGONALIZABLE MATRICES MICHAEL I. GIL Abstract. Let A and à be n n diagonalizable matrices and f be a function defined on their spectra. In the present paper, bounds for the norm of f(a) f(ã) are established. Applications to differential equations are also discussed. Key words. Matrix valued functions, Perturbations, Similarity of matrices, Diagonalizable matrices. AMS subject classifications. 15A54, 15A45, 15A Introduction and statement of the main result. Let C n be a Euclidean spacewiththescalarproduct(, ), Euclideannorm = (, )andidentityoperator I. A and à are n n matrices with eigenvalues λ j and λ j (j = 1,...,n), respectively. σ(a) denotes the spectrum of A, A is the adjoint to A, and N 2 (A) is the Hilbert- Schmidt (Frobenius) norm of A: N 2 2 (A) = Trace(A A). In the sequel, it is assumed that each of the matrices A and à has n linearly independent eigenvectors, and therefore, these matrices are diagonalizable. In other words, the eigenvalues of these matrices are semi-simple. Let f be a scalar function defined on σ(a) σ(ã). The aim of this paper is to establish inequalities for the norm of f(a) f(ã). The literature on perturbations of matrix valued functions is very rich but mainly, perturbations of matrix functions of a complex argument and matrix functions of Hermitian matrices were considered, cf. [1, 11, 13, 14, 16, 18]. The matrix valued functions of a non-hermitian argument have been investigated essentially less, although they are very important for various applications; see the book [10]. The following quantity plays an essential role hereafter: g(a) := [ N 2 2(A) ] 1/2 n λ k 2. Received by the editors December 8, Accepted for publication April 19, Handling Editor: Peter Lancaster. Department of Mathematics, Ben Gurion University of the Negev, P.0. Box 653, Beer-Sheva 84105, Israel (gilmi@cs.bgu.ac.il). Supported by the Kameah fund of the Israel. 303

3 304 M.I. Gil g(a) enjoys the following properties: g 2 (A) 2N 2 2(A I ) (A I = (A A )/2i) and g 2 (A) N 2 2(A) Trace(A 2 ), (1.1) cf. [8, Section 2.1]. If A is normal, then g(a) = 0. Denote by µ j, j = 1,...,m n, the distinct eigenvalues of A, and by p j the algebraic multiplicity of µ j. In particular, one can write µ 1 = λ 1 = = λ p1, µ 2 = λ p1+1 = = λ p1+p 2, etc. Similarly, µ j, j = 1,..., m n, are the distinct eigenvalues of Ã, and p j denotes the algebraic multiplicity of µ j. Let δ j be the half-distance from µ j to the other eigenvalues of A, namely, Similarly, Put β(a) := δ j := δ j := p j min µ j µ k /2 > 0.,...,m; k j min µ j µ k /2 > 0.,..., m; k j δ k k=0 j g k (A) and β(ã) := m p j Here g 0 (A) = δj 0 = 1 and g0 (A) = δ j 0 = 1. According to (1.1), In what follows, we put β(a) p j k=0 ( 2N 2 (A I )) k. δ k j f(λ k ) f( λ j ) λ k λ j = 0 if λ k = λ j. Now we are in a position to formulate our main result. k=0 g k (Ã). δ j k Theorem 1.1. Let A and à be n n diagonalizable matrices and f be a function defined on σ(a) σ(ã). Then the inequalities f(λ k ) f( λ j ) N 2 (f(a) f(ã)) β(a)β(ã)max j,k λ k λ N2(A Ã) (1.2) j and are valid. N 2 (f(a) f(ã)) β(a)β(ã)max f(λ k) f( λ j ) (1.3) j,k The proof of this theorem is divided into lemmas which are presented in the next two sections. The importance of Theorem 1.1 lies in the fact that the right-hand sides of inequalities (1.2) and (1.3) only involve universal quantities calculated for A and Ã, and the values of the function f on the spectra σ(a) and σ(ã), but e.g. no matrices performing similarities of A and à to diagonal ones.

4 Perturbations of Functions of Diagonalizable Matrices The basic lemma. Let T and T be the invertible matrices performing similarities of A and à to diagonal ones, that is, and T 1 AT = S, (2.1) T 1 à T = S, (2.2) where S = diag(λ j ) and S = diag( λ j ). Everywhere below, {d k } n is the standard orthonormal basis and B = sup { Bh / h : h C n } for an n n matrix B. Recall that N 2 (AB) B N 2 (A). We begin with the following lemma. Lemma 2.1. Let the hypotheses of Theorem 1.1 hold. Then N 2 2 (f(a) f(ã)) T 1 2 T 2 n j, (f(λ k ) f( λ j ))(T 1 Tdj,d k ) 2. (2.3) Proof. We have N 2 2 (A Ã) = N2 2 (TST 1 T S T 1 ) = N 2 2 (TST 1 T T 1 TT 1 T S T 1 ) = N2 2 (T(ST 1 T T 1 T S) T 1 ) T 1 2 T 2 N2 2 (ST 1 T T 1 T S). But N2(ST 2 1 T T 1 T S) n = (ST 1 T T 1 T S)dj,d k ) 2 j, n = (T 1 Tdj,S d k ) (T 1 T Sdj,d k ) 2 = J, (2.4) j, where n J := (λ k λ j )(T 1 Tdj,d k ) 2. j, So, N 2(A Ã) 2 T 1 2 T 2 J.

5 306 M.I. Gil Similarly, taking into account that T 1 f(a)t = f(s) = diag (f(λ j )) and T 1 f(ã) T = f( S) = diag (f( λ j )), we get (2.3), as claimed. From the previous lemma we obtain at once: Corollary 2.2. Under the hypotheses of Theorem 1.1, we have n N 2 (f(a) f(ã)) κ T κ T [ f(λ k ) f( λ j ) 2 ] 1/2, j, where κ T := T T 1 and κ T := T T 1. Recall that f(λ k ) f( λ j ) λ k λ j = 0 if λ k = λ j. Note that (2.3) implies N 2 2 (f(a) f(ã)) T 1 2 T 2 n It follows from (2.4) that Thus, (2.5) yields: is true. j, f(λ k) f( λ j ) λ k λ j (λ k λ j )(T 1 Tdj,d k ) 2 T 1 2 T 2 max f(λ k) f( λ j ) 2 J. (2.5) j,k λ k λ j J = N 2 2 (T 1 A T T 1 Ã T) T 1 2 T 2 N 2 2 (A Ã). Corollary 2.3. Under the hypotheses of Theorem 1.1, the inequality N 2 (f(a) f(ã)) κ T κ T N 2 (A Ã)max f(λ k) f( λ j ) j,k λ k λ (2.6) j Since f(λ k) f( λ j ) sup λ k λ f (s), j s co(a,ã)

6 Perturbations of Functions of Diagonalizable Matrices 307 where co(a,ã) is the closed convex hull of the eigenvalues of the both matrices A and Ã, cf. [17], we have: Corollary 2.4. Let A and à be n n diagonalizable matrices and f be a function defined and differentiable on co(a,ã). Then N 2 (f(a) f(ã)) κ T κ T max f (s) N 2 (A Ã). s co(a,ã) We need also the following corollary of Lemma 2.1. Corollary 2.5. Under the hypotheses of Theorem 1.1, it holds that n N 2 (f(a) f(ã)) T 1 T max f(λ n k) f( λ j ) [ Td j 2 (T 1 ) d k 2 ] 1/2. j,k (2.7) 3. Proof of Theorem 1.1. Let Q j be the Riesz projection for µ j : Q j = 1 R λ (A)dλ, (3.1) 2πi L j where L j := {z C : z µ j = δ j }. Lemma 3.1. The inequality is true. Q j k=0 g k (A) δ k j Proof. By the Schur theorem, there is an orthonormal basis {e k }, in which A = D + V, where D = diag (λ j ) is a normal matrix (the diagonal part of A) and V is an upper triangular nilpotent matrix (the nilpotent part of A). Let P j be the eigenprojection of D corresponding to µ j. Then by (3.1), Q j P j = 1 2πi since V = A D. But Consequently, L j (R λ (A) R λ (D))dλ = 1 2πi L j R λ (A)VR λ (D)dλ R λ (A) = (D +V Iλ) 1 = (I +R λ (D)V) 1 R λ (D). R λ (A) = ( 1) k (R λ (D)V) k R λ (D). k=0

7 308 M.I. Gil So, we have where By Theorem of [8], we get Q j P j = C k, (3.2) C k = ( 1) k 1 (R λ (D)V) k R λ (D)dλ. 2πi L j (R λ (D)V) k Nk 2 (R λ(d)v) Nk 2 (V) ρ k (A,λ), where ρ(a,λ) = min k λ λ k. In addition, directly from the definition of g(a), when the Hilbert-Schmidt norm is calculated in the basis {e k }, we have N 2 (V) = g(a). So, and therefore, Hence, C k 1 2π (R λ (D)V) k gk (A) (λ L δj k j ), (R λ (D)V) k R λ (D) dλ gk (A) 1 L j δ k+1 2π j dλ = gk (A). L j δj k and therefore, as claimed. Q j P j C k Q j P j + Q j P j δ k j g k (A), (3.3) δ k k=0 j g k (A), Let {v k } n be a sequence of the eigenvectors of A, and {u k} n be the biorthogonal sequence: (v j,u k ) = 0 (j k), (v j,u j ) = 1 (j,k = 1,...,n). So n A = µ k Q k = λ k (,u k )v k. (3.4) Rearranging these biorthogonal sequences, we can write p j Q j = (,u jk )v jk,

8 Perturbations of Functions of Diagonalizable Matrices 309 where {u jk } pj and {v jk} pj are biorthogonal: (v jk,u jm ) = 0 (m k), (v jk,u jk ) = 1(m,k = 1,...,p j ). Observethatwecanalwayschoosethesesystemssothat u jk = v jk. Lemma 3.2. Let u jk = v jk. Then we have u jk 2 Q j (k = 1,...,p j ). Proof. Clearly, Q j u jk = (Q j u jk,u jk )v jk. So, (Q j u jk,v jk ) = (u jk,u jk )(v jk,v jk ) = u jk 4. Hence, u jk 4 Q j u jk 2, as claimed. Lemmas 3.1 and 3.2 imply: Corollary 3.3. The inequalities are valid. u js 2 δ k k=0 j g k (A) (s = 1,2,...,p j ) Again, let {d k } be the standard orthonormal basis. Lemma 3.4. Let A be an n n diagonalizable matrix. Then the operator has the inverse one defined by n T = (,d k )v k (3.5) n T 1 = (,u k )d k, (3.6) and (2.1) holds. Proof. Indeed, we can write out n n n T 1 T = d j (,d k )(v k,u j ) = d j (,d j ) = I and n n n TT 1 = (,u k ) (d k,d j )v j = (,u k )v k = I.

9 310 M.I. Gil Moreover, n n AT = v j λ j (,d j ) and S = λ k (,d k )d k. Hence, n n n T 1 AT = d k (v j,u k )λ j (,d j ) = d j λ j (,d j ) = S. So, (2.1) really holds. Lemma 3.5. Let T be defined by (3.5). Then T 2 p j Q j and T 1 2 and therefore, κ T p j Q j. p j Q j, Proof. Due to the previous lemma T 1 x 2 = By the Schwarz inequality, n n (x,u k ) 2 x 2 u k 2 (x C n ). (3.7) s=1 n n (Tx,Tx) = (x,d k )(v k,v s )(x,d s ) n n n (x,d k ) 2 [ (v k,v s ) 2 ] 1/2 x 2 v s 2. (3.8) s, s=1 But by Lemma 3.2, p n n j u s 2 = v s 2 = v jk 2 p j Q j. s=1 s=1 Now (3.7) and (3.8) yield the required result. Lemmas 3.1 and 3.5 imply:

10 Perturbations of Functions of Diagonalizable Matrices 311 Corollary 3.6. The inequality κ T β(a) is true. Lemma 3.7. Under the hypotheses of Theorem 1.1, we have N 2 (f(a) f(ã)) T 1 T max j,k f(λ k) f( λ j ) [ p j Q j where Q k, Q j are the eigenprojections of A and Ã, respectively. Proof. By (3.6), we have Moreover, due to Lemma 3.2, Similarly, (T 1 ) = n (T 1 ) d j 2 = Now (2.7) implies the required result. n (,d k )u k. n u k 2 p k Q k. n Td k 2 p k Q k. Thanks to Lemmas 3.5 and 3.7 we get the inequality N 2 (f(a) f(ã)) max f(λ k) f( λ j ) j,k p j Q j p k Q k ] 1/2, p k Q k. (3.9) Proof of Theorem 1.1: Inequality (1.2) is due to (2.6) and Corollary 3.6. Inequality (1.3) is due to (3.9) and Lemma Applications. Let us consider the two differential equations du dt = Au (t > 0) and dũ dt = Ãũ (t > 0), whose solutions are u(t) and ũ(t), respectively. Let us take the initial conditions u(0) = ũ(0) C n. Then we have By (1.2), u(t) ũ(t) = (exp [At] exp [Ãt])u(0). u(t) ũ(t) u(0) N 2 (exp [At] exp [Ãt])

11 312 M.I. Gil where te tα N 2 (A Ã)β(A)β(Ã) u(0) (t 0), α = max{max k Re λ k,maxre λ j }. j Furthermore, let 0 σ(a) σ(a). Then the function A 1/2 sin (A 1/2 t) (t > 0) is the Green function to the Cauchy problem for the equation Simultaneously, consider the equation and take the initial conditions Then we have d 2 w +Aw = 0 (t > 0). dt2 d 2 w +Ã w = 0 (t > 0), dt2 w(0) = w(0) C n and w (0) = w (0) = 0. w(t) w(t) = [A 1/2 sin(a 1/2 t) Ã 1/2 sin(ã1/2 t)]w(0). By (1.2), w(t) w(t) β(a)β(ã)n 2(A Ã)max j,k sin (t λ k ) λk sin (t λj) λj λ k λ j (t > 0). Here one can take an arbitrary branch of the roots. Acknowledgment. I am very grateful to the referee of this paper for his really helpful remarks. REFERENCES [1] R. Bhatia. Matrix Analysis. Springer, Berlin, [2] K.N. Boyadzhiev. Some inequalities for generalized commutators. Publ. Res. Inst. Math. Sci., 26(3): , 1990.

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