R. Bhatia. Perturbation Bounds for Matrix Eigenvalues. Pitman Research Notes in Mathematics. Longmann Scientific & Technical, Harlow, Essex, 1987. Published in the USA by John Wiley.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....to those of A. One may ask now, which of the three variants examined produces the best approximation in terms of smallest eigenvalue variation. We use the criterion j i (A(t 2 ) Gamma i (A(t 1 ) j j i (A(t 2 ) Gamma A(t 1 ) j; which is derived from Theorem 9. 7 in Bhatia [14] with the Ky Fan n norm [14, p.28] Table 3.1 summarizes the upper bounds for all three variants. The rank one and modification bound rank 1 modification 2(t 2 Gamma t 1 )jfi k j rank 2 2(t 2 Gamma t 1 )jfi k j rank 1 extension 2(t 2 Gamma t 1 ) Table 3.1: theoretical bound for j ....

R. Bhatia. Perturbation bounds for matrix eigenvalues. Longman Scientific and Technical, Harlow, England, 1987.

....use of Weyl s Monotonicity Principle (see e.g. 8, Corollary IV.4.9, and the subsequent discussion] or [4, Corollary 4.3.3] If X Y are k Theta k then i (X) i (Y ) i = 1; 2; k: We will also need the Lidskii Wielandt bound which is the next result. See [8, Theorem IV.4. 8] or [1] for the traditional proof via Wielandt s min max Theorem or see [5] for a much more elementary proof using Weyl s monotonicity principle. Theorem 2 Let X and Y be n Theta n Hermitian matrices, and let Phi be a symmetric gauge function on R n then Phi( 1 (X) Gamma 1 (Y ) n (X) ....

R. Bhatia. Perturbation Bounds for Matrix Eigenvalues. Pitman Research Notes in Mathematics 162. Longman Scientific and Technical, New York, 1987.

....use of Weyl s Monotonicity Principle (see e.g. 10, Corollary IV.4.9, and the subsequent discussion] or [5, Corollary 4.3.3] If X Y are k Theta k then i (X) i (Y ) i = 1; 2; k: 4 We will also need the Lidskii Wielandt bound which is the next result. See [10, Theorem IV.4. 8] or [1] for the traditional proof via Wielandt s min max Theorem or see [6] for a much more elementary proof using Weyl s monotonicity principle. Theorem 2 Let X and Y be n Theta n Hermitian matrices, and let Phi be a symmetric gauge function on R n then Phi( 1 (X) Gamma 1 (Y ) n (X) ....

R. Bhatia. Perturbation Bounds for Matrix Eigenvalues. Pitman Research Notes in Mathematics 162. Longman Scientific and Technical, New York, 1987.

....p is a metric on C is open. 4 Known Perturbation Theorems for Eigenvalue and Singular Value Variations In this section, we will briefly review several most celebrated theorems for eigenvalue and singular value variations which will be extended later. Most of these theorems can be found in Bhatia [3, 1987], Golub and Van Loan [14, 1989] Parlett [33, 1980] and Stewart and Sun [35, 1990] Notation introduced at the beginning of x2 will be followed strictly. Hoffman and Wielandt [16, 1953] proved Theorem 4.1 (Hoffman Wielandt) If A and e A are normal, then there is a permutation of f1; 2; Delta ....

R. Bhatia. Perturbation Bounds for Matrix Eigenvalues. Pitman Research Notes in Mathematics. Longmann Scientific & Technical, Harlow, Essex, 1987. Published in the USA by John Wiley. Ren-Cang Li: Relative Perturbation Theory 50 b g y=f(x) is convex. a b g a y=g(x) is concave.

.... e i2 ) 1 e i5 ) det[ T E) Gamma I] 1 fl) where 1 fl = n Y i=1 (1 e i1 ) 1 e i2 ) 1 e i5 ) with e 12 = e 15 = 0 and je i5 j , i = 2; Delta Delta Delta ; n. It can be easily seen that Gamma(3n Gamma 2) O( 2 ) fl (3n Gamma 2) O( 2 ) By Weyl s Inequality ([3] p34, or Lemma 6.1 in x6.1) if 1 2 Delta Delta Delta n are zeros of f( j (1 fl) det[ T E) Gamma ] then j i Gamma i j kEk 2 2:5 max j (jfi j j jfi j 1 j) O( 2 ) i = 1; 2; Delta Delta Delta ; n 8 T. Y. LI AND Z. ZENG where i s are eigenvalues of T ....

....p462 ) However, if we consider the distance from the eigenvalues of perturbed matrices T (1) or T (2) to the corresponding eigenvalues of T , our rank two tearing is better than the rank one tearing, because we can always apply Laguerre s iteration from closer starting points. Lemma 6. 1 ([3] p34) Let A and B be real symmetric matrices with eigenvalues 1 (A) Delta Delta Delta n (A) and 1 (B) Delta Delta Delta n (B) respectively. Then max j j j (A) Gamma j (B)j kA Gamma Bk: Lemma 6.2. For real symmetric matrices A and B, let f i (A)g n i=1 , f i (B)g n i=1 and f i ....

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R. Bhatia, Perturbation Bounds for Matrix Eigenvalues, John Wiley & and Sons, Inc, New York, 1987.

....result we therefore need to show any matrix G in S n satisfies Pi m (0; G) 0 1 (0; G) or, using formula (2.2) sup H2S n fm (H G) Gamma m (H)g = 1 (G) 4.3) On the one hand, the inequality m (H G) m (H) 1 (G) is a classical inequality of Weyl (see [12, Thm 4.3. 7] or [1]) To see the supremum is attained, choose an orthogonal matrix U with U T GU = Diag (G) and any real ff 1 (G) Gamma n (G) Then if we set H = U Diag (0; 0; 0; ff; ff; ff z m Gamma1 terms ) U T we deduce m (H G) Gamma m (H) m (U T (H G)U) Gamma 0 = m ....

R. Bhatia. Perturbation bounds for matrix eigenvalues. Research Notes in Mathematics 162. Pitman, 1987.

....j = 1; n, and hence k X j=1 [ i j (A E ) Gamma i j (A) n X j=1 [ j (A E ) Gamma j (A) trace(A E ) Gamma trace(A) trace(E ) as desired. 2 Typically one derives the Lidskii Mirsky Wielandt bound from Wielandt s min max representation of P k j=1 i j (A) [2] [19] This is considerably more complicated than our approach. 1 In the last section we show that our technique can be used to prove Wielandt s min max theorem. Now we use the same technique to prove a multiplicative analog of (1.1) Theorem 2.3 Let A be an n Theta n Hermitian matrix and let ....

R. Bhatia. Perturbation Bounds for Matrix Eigenvalues. Pitman Research Notes in Mathematics 162. Longman Scientific and Technical, New York, 1987.

....result we therefore need to show any matrix D in S n satisfies Pi m (0; D) 0 1 (0; D) or, using formula (2.2) sup H2S n fm (H D) Gamma m (H)g = 1 (D) 4.3) On the one hand, the inequality m (H D) m (H) 1 (D) is a classical inequality of Weyl (see [10, Thm 4.3. 7] or [1]) To see the supremum is attained, choose an orthogonal matrix U with U T DU = Diag (D) and any real ff 1 (D) Gamma n (D) Then if we set H = U Diag (0; 0; 0; ff; ff; ff z m Gamma1 terms ) U T we deduce m (H D) Gamma m (H) m (U T (H D)U) Gamma 0 = m ....

R. Bhatia. Perturbation bounds for matrix eigenvalues. Research Notes in Mathematics 162. Pitman, 1987.

.... Gamma1 k 2 2 (Theorem 5.3) 4 Known Perturbation Theorems for Eigenvalue and Singular Value Variations In this section, we will briefly review a few most celebrated theorems for eigenvalue and singular value variations which will be generalized. Most of this theorems can be found in Bhatia [3], Golub and Van Loan [14] Parlett [28] and Stewart and Sun [30] Notation introduced in x3 will be followed strictly. Hoffman and Wielandt [16] proved Theorem 4.1 (Hoffman Wielandt) If A and e A are normal, then there is a permutation of f1; 2; Delta Delta Delta ; ng such that v u u t n ....

....and as a corollary of Hoffman Wielandt theorem [16] for the Frobenius norm and by Lidskii [23] Wielandt [36] and Mirsky [27] for all unitarily invariant norms. Neither Lidskii nor Wielandt mentioned explicitly (4. 3) which was done by Mirsky [27] For more detail, the reader is referred to Bhatia [3]. Theorem 4.3 has been generalized in many aspects. The following theorem is due to Bhatia, Davis and Kittaneh [4] Theorem 4.4 (Kahan, Bhatia, Davis and Kittaneh) To the hypotheses of Theorem 4.2 adds this: all i s and e j s are real and are arranged descendingly as in (3.4) Then for any ....

R. Bhatia. Perturbation Bounds for Matrix Eigenvalues. Pitman Research Notes in Mathematics. Longmann Scientific & Technical, Harlow, Essex, 1987. Published in the USA by John Wiley.

....ffi 0. Consider first METRIC ENTROPY OF HOMOGENEOUS SPACES 7 the case of the operator norm. We need to show that if A; B are Hermitian with spectra contained in [ Gamma ; and ke iA Gamma e iB k ffi, then kA Gamma Bk C( ffi, where C( depends only on (and not on A; B or n) By [1], Theorem 13.6, the eigevalues of A and B (multiplicities counted) are, ia a certain precise sense, close , and so, by perturbation, we may assume that they are identical; we may also assume that all those eigenvalues are inegral multiples of ffi. Let u 2 U(n) be such that B = uAu Gamma1 ; we ....

....sphere fx : kxk = 1g does not contain a segment) iff the associated symmetric norm on R n is (cf. 7] The operator norm and the trace class C 1 norm are not strictly convex. 2) It is a somewhat delicate issue how smooth should be the curves we consider, particularly since the results from [1], to which we refer quite heavily, do not fit precisely our needs exactly in that respect. For the purpose of following the proof below, the reader should think of all functions as being at least C 1 . As indicated in the previous section, absolutely continuous functions provide a convenient ....

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R. Bhatia, Perturbation bounds for matrix eigenvalues, Pitman Research Notes 162, Longman Scientific & Technical, Harlow 1987.

....for every unitarily invariant norm. Proof: Note that jjjA Gamma Bjjj = jjjS S Gamma1 Gamma TMT Gamma1 jjj = jjjS (S Gamma1 T Gamma S Gamma1 TM)T Gamma1 jjj kSk kT Gamma1 k jjjS Gamma1 T Gamma S Gamma1 TM jjj; 4) by a familiar property of unitarily invariant norms [2]. So, to prove (3) it would suffice to show that jjjX Gamma XM jjj kXk jjj # Gamma M jjj; 5) for every matrix X. This is equivalent to the assertion jjjX Gamma XM jjj jjj # Gamma M jjj; 6) for every matrix X with kXk = 1. It is an easy consequence of the singular value ....

R. Bhatia, Perturbation Bounds for Matrix Eigenvalues, Longman, Essex and Wiley, New York, 1987.

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R. Bhatia. Perturbation Bounds for Matrix Eigenvalues. Pitman Research Notes in Mathematics. Longmann Scientific & Technical, Harlow, Essex, 1987. Published in the USA by John Wiley.

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R. Bhatia. Perturbation Bounds for Matrix Eigenvalues. Pitman Research Notes in Mathematics 162. Longman Scientific and Technical, New York, 1987.

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R. Bhatia. Perturbation Bounds for Matrix Eigenvalues. Pitman Research Notes in Mathematics 162. Longman Scientific and Technical, New York, 1987.

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R. Bhatia. Perturbation Bounds for Matrix Eigenvalues. Pitman Research Notes in Mathematics 162. Longman Scientific and Technical, New York, 1987.

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