Identical pseudospectra of any geometric multiplicity

Identical pseudospectra of any geometric multiplicity∗ Gorka Armentia† , Juan-Miguel Gracia‡ , Francisco E. Velasco‡ January 9, 2011 Dedicated to Professor Jos´e Ant´onio Dias da Silva Abstract If A, B are n×ncomplex matrices such that the singular values of zIn−Aare the same as those of zIn−Bfor each z∈C, then Aand B are similar. AMS classification: 15A18, 15A21, 15A60, 47A25. Key Words: singular values, similarity, pseudospectrum, infinitesimals. 1 Introduction Let M∈Cn×n. Let Λ(M) denote the spectrum of Mand let σ1(M)≥ σ2(M)≥ · · · ≥ σn(M) denote the singular values of Marranged in decreasing order. We write k·k2for the Euclidean norm on Cn, defined by kxk2:= ( P n i=1 |xi|2)1/2, and k·kfor the associated operator norm on Cn×n, defined by kMk:= sup{kMxk2:kxk2= 1}. We write GLn(C) for the group of invertible matrices of Cn×n. Given ε≥0, the ordinary ε-pseudospectrum of Mcan be defined as the set Λε(M) := {z∈C:σn(zIn−M)≤ε}. In the Ph.D. Thesis of M. Karow [4] the relationship was shown between the condition numbers of eigenvalues of a matrix M∈Cn×n, whose spectrum is {λ1, . . . , λp}, and its Jordan decomposition M= p X i=1 (λiPi+Ni), where Piis the Riesz projector corresponding to λiand Niis the eigennilpotent matrix associated with λi. In particular, the index ν(λi) of each eigenvalue λi plays a major role. Moreover, in the same dissertation, the condition number of the eigenvalue λiis related to the connected component of the pseudospectrum ∗This work was supported by the Ministry of Education and Science, Project MTM 200767812-CO2-01. †Department of Mathematical Engineering and Computer Science, The Public University of Navarre, Campus de Arrosad´ıa, 31006 Pamplona, Spain. [email protected] ‡Department of Applied Mathematics and Statistics, The University of the Basque Country, Faculty of Pharmacy, 7 Paseo de la Universidad, 01006 Vitoria-Gasteiz, Spain, [email protected], [email protected] 1 This is the accepted manuscript of the article that appeared in final form in Linear Algebra and its Applications 436(6) : 1683-1688 (20212), which has been published in final form at https://doi.org/10.1016/ j.laa.2011.01.014. © 2011 Elsevier under CC BY-NC-ND license (http://creativecommons.org/licenses/by-ncnd/4.0/) Λε(M) containing λi. These facts led us to think that there should be a closer relationship between the Jordan canonical form of Aand its pseudospectra. Let kbe an integer, 1 ≤k≤n. For ε≥0, the geometric ε-pseudospectrum of Mof order kcan be defined as the set Λ(g) ε,k(M) = {z∈C:σn−k+1(zIn−M)≤ε}. In this paper we are going to establish that the geometric pseudospectra Λ(g) ε,k(A) for small enough εdetermine the Jordan canonical form of A, or equivalently, determine its invariant factors. This is the content of the main theorem in this paper, which is the following. Theorem 1 (Sufficient condition for similarity).Let A, B ∈Cn×n. Let us assume that for each z∈Cthe singular values of zIn−Aare the same as those of zIn−B. Then Aand Bare similar matrices. This Theorem will be proved in Section 3. Remark 1. Notice that if Aand Bare similar and both matrices are normal, then for each z∈Cthe singular values of zIn−Aare the same as those of zIn−B. This is no longer true if Aand Bare not assumed normal. Theorem 1 was also inspired by Fact 5(b), page 16-2 in Chapter 16 on Pseudospectra written by M. Embree in the Handbook of Linear Algebra, edited by L. Hogben [3]. This Fact says that if Aand Bare n×ncomplex matrices that have the same ordinary ε-pseudospectrum for every ε > 0, then Aand B have the same minimal polynomial. We remark that Λε(M) = Λ(g) ε,1(M). Once we had proven our theorems, we read the paper by M. Fortier Bourque and T. Ransford [1], which came to confirm our hunch. Two matrices A, B ∈ Cn×nare said to be unitarily similar if there exists a unitary matrix U∈Cn×n such that B=U∗AU, where ∗stands for the conjugate transpose. M. F. Bourque and T. Ransford say that the complex n×nmatrices Aand Bhave super-identical pseudospectra if, for each z∈C, the singular values of zIn−A are the same as those of zIn−B. In [1] it was also proved that this condition is excessive, and it is sufficient to require these equalities for a certain finite set Fof C; namely, Theorem 2. Let F:= {rpeiθq:p, q = 0, . . . , n}, where 0< θ0<· · · < θn< π and 0< r0<· · · < rn. Suppose that A, B ∈Cn×nsatisfy σk(zIn−A) = σk(zIn−B) (z∈F, k = 1, . . . , n). Then Aand Bhave super-identical pseudospectra. Also they showed that: (a) A, B ∈C2×2have super-identical pseudospectra if and only if Ais unitarily similar to B; (b) A, B ∈C3×3have super-identical pseudospectra if and only if Ais unitarily similar to Bor to its transpose; (c) there exist A, B ∈C4×4with super-identical pseudospectra such that kA2k 6= kB2k, this implies that Ais not unitarily similar either to Bor to its transpose. We would note that there are problems in pure mathematics and control theory where the simultaneous consideration of all the singular values leads to more satisfactory solutions, like the problem of studying the approximation 2 of a bounded matrix function on the unit circle by bounded analytic matrix functions on the unit disc [5]. The organization of this paper is as follows: Given M∈Cn×nand z0an eigenvalue of M, we will analyze the asymptotic behavior of the singular values of the characteristic matrix zIn−Mwhen z→z0in Section 2. We will prove Theorem 1 in Section 3. In Section 4 we will give an extension of Theorem 1, and we will frame these results in the theory of pseudospectra. 2 Orders of the singular values of a characteristic matrix as infinitesimals Let a matrix M∈Cn×nand z0an eigenvalue of M. In this section we will study the asymptotic behavior of the singular values of the characteristic matrix zIn−M, when z→z0. To that end, we need the following notations. Let V0(z0) be a punctured neighborhood of z0in C, we consider the set Fof real functions defined on V0(z0). Then, we have the following definition. Definition 1. Let f, g ∈F. If there are constants δ, ∆, d > 0 such that for every z∈B0(z0, d) (open punctured disk centered at z0and radius d) f(z)>0, g(z)>0 and δ≤f(z) g(z)≤∆, we write (with Hardy’s notation [2]) f(z)g(z) (when z→z0). We say that a function f∈Fis an infinitesimal as z→z0if limz→z0f(z) = 0. If f(z) |z−z0|k(with kinteger ≥1) we say that f(z) is an infinitesimal of order kas z→z0. The relation is an equivalence relation. Remark 2. If j, k are integers ≥0 and |z−z0|j |z−z0|k(z→z0), then j=k. Remark 3. Recall that for positive functions f, g ∈Fthe relation f(z)∼g(z) as z→z0means lim z→z0 f(z) g(z)= 1. It is obvious that f(z)∼g(z) as z→z0implies f(z)g(z) as z→z0. The main result of this section is the following lemma. Lemma 3. If Jk(z0)is the k×kJordan block with eigenvalue z0, then, as z→z0, σj  zIk−Jk(z0)  ∼ ¨ 1, j = 1, . . . , k −1, |z−z0|k, j =k. 3 Proof. Without loss of generality, we may suppose that z0= 0 and write simply Jk:= Jk(0). Since J∗ kJk= diag(0,1,...,1), it follows that the singular values of Jkare 1,...,1,0. Hence σj(zIk−Jk)→1 as z→0 for j= 1,2, . . . , k −1. Also k Y j=1 σj(zIk−Jk)2= det((zIk−Jk)∗(zIk−Jk)) = |det(zIk−Jk)|2=|z|2k, whence it follows that σk(zIk−Jk)∼ |z|kas z→0. 2 For the proof of Lemma 7, we need some preliminary results. The first one can be seen in [6]. Lemma 4. Let M1, M2, M3∈Cn×n. Then, for k= 1,2, . . . , n, σn(M1)σk(M2)σn(M3)≤σk(M1M2M3)≤ kM1kkM3kσk(M2). With this result we can prove the following. Lemma 5. Let M∈Cn×n, P ∈GLn(C)and z0∈C. Then, for j= 1,2, . . . , n, σj(zIn−P−1MP)σj(zIn−M) (z→z0). Lemma 6. Let L∈Cq×qand z0be a complex number such that z0/∈Λ(L). Then, for j= 1,2, . . . , q, σj(zIq−L)1 (z→z0). Proof. For j= 1,2, . . . , q, the limit lim z→z0 σj(zIq−L) = σj(z0Iq−L) is nonzero and finite. 2 Lemma 7. Let Jbe the Jordan form of a matrix M∈Cn×n. Let z0∈Cand k∈ {1, . . . , n}. Then the number of k×kJordan blocks in Jwith eigenvalue z0 is equal to the number of j∈ {1, . . . , n}such that σj(zIn−M) |z−z0|kas z→z0. Proof. By Lemma 6 if z0/∈Λ(M), then for j∈ {1, . . . , n}, σj(zIn−M)1 (z→z0); so, in this case, there is no jsuch that σj(zIn−M) |z−z0|kas z→z0. If z0∈Λ(M), by Lemma 5, for j= 1, . . . , n, σj(zIn−M)σj(zIn−J) (z→z0). Let J=J0⊕J1, 4 where J0∈Cn0×n0is the direct sum of the tJordan blocks associated with z0, and z0/∈Λ(J1). When zis sufficiently close to z0, the last tsingular values of zIn−Jare just the infinitesimal singular values of zIn0−J0as z→z0. Thus, lim z→z0 σj(zIn−J)=0,for j=n−t+ 1, . . . , n −1, n. The number of j∈ {n−t+1, . . . , n−1, n}such that the order of the infinitesimal σj(zIn−J) as z→z0is k, is equal to the number of k×kJordan blocks in J0 associated with z0. For j∈ {1, . . . , n −t}, we have σj(zIn−J)1 as z→z0. 2 3 Proof of the main result In this section, we will prove the main result of this paper. Proof of Theorem 1. Let M∈Cn×nand z0∈C. Then z0∈Λ(M) if and only if σn(z0In−M) = 0. Since for each z∈C,σn(zIn−A) = σn(zIn−B), the eigenvalues of Aand B are the same, Λ(A) = Λ(B) = {λ1, λ2, . . . , λp}. As σj(zIn−A) = σj(zIn−B) for z∈Cand j∈ {1, . . . , n}, then for each k∈ {1, . . . , n}and λi∈Λ(A), the number of j∈ {1, . . . , n}such that σj(zIn−A) |z−λi|kas z→λi is equal to the number of j∈ {1, . . . , n}such that σj(zIn−B) |z−λi|kas z→λi. Thus, by Lemma 7, the number of k×kJordan blocks associated with λiin the Jordan forms of Aand Bis the same. Given that this holds for every λi∈Λ(A) = Λ(B), we infer that Aand Bare similar. 2 4 Remarks Following a line of reasoning similar to that of Theorem 1, we can establish the following theorem. Theorem 8. Let A∈Cn×n, B ∈Cm×m. Let us suppose that n≥mand let gi(λ)|gi+1(λ)|···|gm−1(λ)|gm(λ) be the nontrivial invariant factors of B. Let us assume that for each z∈Cand k= 1,2, . . . , m −i+ 1, σn−k+1(zIn−A) = σm−k+1(zIm−B) (1) Then the last m−i+ 1 invariant factors of A, fn−m+i(λ)|fn−m+i+1(λ)|···|fn−1(λ)|fn(λ), 5 are nontrivial, and fn(λ) = gm(λ), fn−1(λ) = gm−1(λ), . . . , fn−m+i(λ) = gi(λ). Let M∈Cn×n. For every real number ε≥0, another equivalent definition of the ordinary ε-pseudospectrum of Mis Λε(M) := [ X∈Cn×n kX−Mk≤ε Λ(X). For z∈Cwe denote by gm(z, M) the geometric multiplicity of zas eigenvalue of M. If z /∈Λ(M), we agree that gm(z, M) = 0. Let kbe an integer, 1 ≤k≤n, and let Λ(g) k(M) denote the set of z∈Λ(M) such that gm(z, M)≥k. For ε≥0, the geometric ε-pseudospectrum of Mof order kcan be defined, alternatively, by Λ(g) ε,k(M) := [ X∈Cn×n kX−Mk≤ε Λ(g) k(X). 5 Conclusions Let A, B be n×ncomplex matrices such that the singular values of zIn−Aare the same as those of zIn−Bfor each z∈C. Then Aand Bare similar. A more general result for square matrices Aand Bof distinct sizes has been stated. Acknowledgement We thank the referee for the criticisms, suggestions and the proof of Lemma 3. References [1] M. Fortier Bourque, T. Ransford. Super-identical pseudospectra, J. London Math. Soc. (2) 79 (2009) 511–528. [2] G.H. Hardy. Orders of infinity. Hafner Publishing Company, New York, 1971. [3] L. Hogben. Handbook of Linear Algebra. Chapman, Hall/CRC, 2007. [4] M. Karow. Geometry of spectral value sets. Ph.D. Thesis. University of Bremen, 2003. [5] V.V. Peller, N.J. Young. Superoptimal analytic approximations of matrix functions. J. Funct. Anal. 120 (1994) 300–343. [6] J. F. Queir´o, E. Marques de S´a. Singular values and invariant factors of matrix sums and products. Linear Algebra Appl. 225 (1995)43–56. 6