The Extended Abstracts of The 4th Seminar on Functional Analysis and its Applications 2-3rd March 2016, Ferdowsi University of Mashhad, Iran

ON SPECTRAL AND PSEUDOSPECTRAL RADIUS OF MATRICES

MADJID KHAKSHOUR1∗, GHOLAMREZA AGHAMOLLAEI2, ALEMEH SHEIKH HOSEINI2

1Department of Mathematics, Graduate University of Advanced Technology of Kerman, Kerman, Iran [email protected]

2Department of Pure Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran [email protected]; [email protected]

Abstract. In this paper, some results about the pseudospectral radius of square complex matrices, extending the same results for the classic spectral radius, are stated.

1. Introduction

Let Mn be the algebra of all n × n complex matrices equipped with the operator norm k.k induced by the usual vector norm kxk = (x∗x)1/2 on Cn, i.e., n kAk = max{kAxk : x ∈ C , kxk = 1}. The numerical range of A ∈ Mn is defined and denoted by: ∗ n W (A) = {x Ax : x ∈ C , kxk = 1}.

2010 Mathematics Subject Classification. Primary 15A18; Secondary 15A60, 65F15. Key words and phrases. Pseudospectrum, pseudospectral radius, spectral radius, numerical range, numerical radius. ∗ Speaker. 1 2 KHAKSHOUR, AGHAMOLLAEI, SHEIKH HOSEINI

It is known that for every A ∈ Mn,W (A) is a convex and compact subset of the complex plane containing the spectrum of A. The numerical radius of A ∈ Mn is defined by ω(A) = sup{|λ| : λ ∈ W (A)}. Denote the spectrum of A by σ(A) and the spectral radius of A by ρ(A), i.e., ρ(A) = max{|λ| : λ ∈ σ(A)}. For any  > 0 and any matrix A ∈ Mn, the −pseudospectrum, pseudospectrum for short, of A is defined and denoted, e.g., see [4], by −1 σ(A) = {z ∈ C : k(A − zI) k ≥ 1/}. (1.1) It is known that

σ(A) = {z ∈ σ(A + E): E ∈ Mn and kEk≤ } (1.2) = {z ∈ C : sn(zI − A) ≤ }, where sn(.) denotes the smallest singular value. The −pseudospectral radius, pseudospectral radius for short, of A is defined as

ρ(A) := sup{|λ| : λ ∈ σ(A)}. The theory of pseudospectrum and pseudospectral radius provides an analytical and graphical alternative for investigating nonnormal ma- trices and operators, gives a quantitative estimate of departure from non-normality and also gives information about stability; e.g., see [1], [3] and [4], and their references. In this paper, we extend some known algebraic properties of ρ(.) to ρ(.).

2. Main results The results about the spectral radius stated in this section are well- known and can be found in [2]. In this section, we assume that A and B ∈ Mn, and in every theorem, we state a known result about the spec- tral radius and then we state a similar result for the −pseudospectral radius of matrices. Theorem 2.1. ρ(AB) = ρ(BA).

Theorem 2.1. ρ(AB) ≤ ρ+δ(BA), where δ = kAB − BAk.

Theorem 2.2. ρ(A) = lim kAkk1/k. k→∞ k 1/k Theorem 2.2. ρ(A) = lim sup k(A + E) k . k→∞ kEk≤ ON SPECTRAL AND PSEUDOSPECTRAL RADIUS OF MATRICES 3

Theorem 2.3. ρ1/k(Ak) = ρ(A), for all k ∈ N. 1/k k Theorem 2.3. lim ρ (A ) ≥ max{1, ρ(A)}. k→∞

Theorem 2.4. If AB = BA, then ρ(A + B) ≤ ρ(A) + ρ(B).

Theorem 2.4. If AB = BA, then ρ(A + B) ≤ ρ(A) + ρ(B).

Theorem 2.5. If AB = BA, then ρ(AB) ≤ ρ(A)ρ(B).

Theorem 2.5. If A, B∈ / {αI, α ∈ C}, AB = BA and  ≤ ρ(A) + ρ(B) − 1, then ρ(AB) ≤ ρ(A)ρ(B). In next theorem, for every nonsingular matrix S ∈ Mn, k(S) denotes its condition number, i.e., k(S) = kSkkS−1k. Theorem 2.6. Let A = SBS−1. Then ρ(A) = ρ(B). −1 Theorem 2.6. Let A = SBS . Then ρ(A) ≤ ρk(S)(B). Eigenvalues can change dramatically with small perturbations, a warning that analysis based on them can be misleading. The following theorem hints that pseudospectra may be more robust.

Theorem 2.7. Let E ∈ Mn. Then ρ(A + E) ≤ ρkEk(A).

Theorem 2.7. Let E ∈ Mn. Then ρ(A + E) ≤ ρ+kEk(A). Theorem 2.8. ρ(A) ≤ ω(A).

Theorem 2.8. ρ(A) ≤ ω(A) + . The convergence analysis of stationary iterative methods is based on the behavior of powers of the iteration matrix. It has long been known that transient growth can occur even when the spectral radius of the iteration matrix is less than one, e.g., see [5, p. 63]. The following two theorems use pseudospectra to describe this transient growth. Theorem 2.9. If ρ(A) > 1, then sup kAkk = ∞. k>0 + k Theorem 2.9. Let m ∈ R . If ρ(A) > 1 + m, then sup kA k > m. k≥0

Theorem 2.10. ρk(A) ≤ kAkk, for all k ∈ N. k k kkAkk−1 Theorem 2.10. ρ (A) ≤ kA k + k , for all k ∈ N with k < 1− kAk kAk. n X For the following two theorems, define rj = |ajk|, where j = k=1 1, 2, . . . , n. 4 KHAKSHOUR, AGHAMOLLAEI, SHEIKH HOSEINI

Theorem 2.11. ρ(A) ≤ max {rj}. j=1,...,n √ Theorem 2.11. ρ(A) ≤ max {rj} + n. j=1,...,n The next theorem is a modest step in this direction, a precise map- ping theorem for linear transformations [4]. Theorem 2.12. For α, β ∈ C, ρ(αA + β) ≤ |α|ρ(A) + |β|.

Theorem 2.12. For α ∈ C \{0}, β ∈ C, ρ(αA + β) ≤ |α|ρ/|α|(A) + |β|. For the final result, let V denote an n × k rectangular matrix with orthonormal columns, where k ≤ n, as might be obtained by Arnoldi or subspace iteration, and let H denote a k × k square matrix. In the Arnoldi iteration, H would have Hessenberg form, but this is not necessary for these theorems. First, we assume that the columns of V exactly span an invariant subspace of A. The resulting theorem forms the basis for algorithms that compute pseudospectra by projecting A onto a carefully chosen invariant subspace [4]. Theorem 2.13. If AV = VH, then ρ(H) ≤ ρ(A).

Theorem 2.13. If AV = VH, then ρ(H) ≤ ρ(A).

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