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An Envelope for the Spectrum of a Matrix

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An Envelope for the Spectrum of a Matrix An envelope for the spectrum of a matrix Panayiotis J. Psarrakos∗ and Michael J. Tsatsomeros† September 22, 2011 Abstract We introduce and study an envelope-type region (A) in the complex plane that contains the eigenvalues of a given n n complexE matrix A. (A) is the intersection of an infinite number of regions× defined by cubic curves.E The notion and method of construction of (A) extend the notion of the numerical range of A, F (A), which is known to be anE intersection of an infinite number of half-planes; as a consequence, (A) is contained in F (A) and represents an improvement in localizing the spectrumE of A. Keywords: eigenvalue bounds, numerical range, cubic curve. AMS Subject Classifications: 15A18, 15A60. 1 Introduction It is well known that the real part of each eigenvalue of a matrix A Cn×n is ∈ bounded above by the largest eigenvalue, say δ1(A), of the hermitian part of A. As a consequence, the spectrum of A lies in the intersection of all half-planes of the form e−i θ(s + i t) : s,t R with s δ (ei θA) , θ [0, 2π]. In fact, this infinite ∈ ≤ 1 ∈ intersection of half-planes coincides with the numerical range of A, F (A). The purpose of this paper is to improve upon the above spectrum localization result. We will achieve this by replacing the infinite intersection of half-planes by an infinite intersection of regions in the complex plane, which are defined by cubic curves. These cubic curves are obtained by an inequality that all eigenvalues of A must satisfy [1]. The outcome is a localization region for the spectrum of A that is contained in F (A), and it can in fact be quite smaller. We will refer to this region as the cubic envelope (or, for simplicity, just envelope) of A, study its basic properties and compare it to the numerical range F (A). ∗Department of Mathematics, National Technical University of Athens, Zografou Campus, 15780 Athens, Greece ([email protected]). †Mathematics Department, Washington State University, Pullman, WA 99164-3113, USA ([email protected]). 1 In Section 2 we describe some of the basic properties of the numerical range needed in our analysis. Section 3 contains a discussion about the cubic curve studied in [1] that bounds the spectrum, along with an answer (see Theorem 3.2) to a question arising in [1]. In Sections 4 and 5, we define and develop the properties of the envelope of A as a containment region of the spectrum of A. Our results and concepts are illustrated with several examples and figures. 2 Preliminaries on the numerical range Let A Cn×n be an n n complex matrix with spectrum ∈ × σ(A) = λ C : det(λIn A) = 0 , { ∈ − } where In denotes the n n identity matrix. Consider also the hermitian and skew- × hermitian parts of A, H(A) = (A + A∗)/2 and S(A) = (A A∗)/2, respectively, and − let δ (A) δ (A) δn(A) denote the eigenvalues of H(A). The numerical 1 ≥ 2 ≥ ··· ≥ range (also known as the field of values) of A is defined as F (A) = v∗Av C : v Cn with v∗v = 1 ; { ∈ ∈ } it is a compact and convex subset of C that contains σ(A) (see [3] and the references therein). Moreover, the following well-known properties of F (A) hold [3]. (P ) For any A Cn×n, F (AT )= F (A) and F (A∗)= F (A)= F (A). In particular, 1 ∈ if A is a real matrix, then F (A) is symmetric with respect to the real axis. (P ) For any a, b C, F (aA + bIn)= aF (A)+ b. 2 ∈ (P ) For any unitary matrix U Cn×n, F (U ∗AU)= F (A). 3 ∈ (P4) The numerical ranges of the hermitian and skew-hermitian parts of A satisfy F (H(A)) = ReF (A) and F (S(A)) = iImF (A). (P5) If A is normal, then F (A) coincides with the convex hull of σ(A), Co(σ(A)). Moreover, A is hermitian if and only if F (A)= F (H(A)) = [δn(A), δ1(A)]. (P ) For any unit vector v Cn (i.e., v∗v = 1), the following are equivalent: 6 0 ∈ 0 0 (i) Re v∗Av = max Re z : z F (A) , { 0 0} { ∈ } (ii) v∗H(A)v = max h : h F (H(A)) , and 0 0 { ∈ } (iii) H(A)v0 = δ1(A)v0. By Property (P6), we have that max Re z : z F (A) = max s : s F (H(A)) = δ (A), { ∈ } { ∈ } 1 that is, the largest eigenvalue of H(A) coincides with the real part of the right most point of F (A). Furthermore, if y Cn is a unit eigenvector of H(A) corresponding 1 ∈ 2 to δ (A), then the right most point of F (A) is y∗Ay and the vertical line = z 1 1 1 L0 { ∈ C : Re z = δ (A) is tangential to F (A) at y∗Ay . 1 } 1 1 Based on the latter observation, Johnson [4] (see also [3, Section 1.5]) proposed an algorithm for the estimation of the numerical range by computing (and plotting) its boundary points. Specifically, for each angle θ [0, 2π], we consider the largest i θ ∈ eigenvalue δ1(e A) and an associated unit eigenvector y1(θ) of the hermitian matrix i θ ∗ H(e A). Then the point zθ = y1(θ) Ay1(θ) lies on the boundary of F (A), denoted by ∂F (A), and the line −i θ i θ θ = e (δ (e A) + i t) : t R L { 1 ∈ } = t sin θ + δ (ei θA) cos θ + i (t cos θ δ (ei θA)sin θ) : t R { 1 − 1 ∈ } is tangential to F (A) at zθ. Moreover, line θ separates the complex plane into the L closed half-plane i θ −i θ i θ in(e A) = e (s + i t) : s,t R with s δ (e A) , H { ∈ ≤ 1 } which contains F (A), and the open half-plane i θ −i θ i θ out(e A) = e (s + i t) : s,t R with s > δ (e A) . H { ∈ 1 } Thus, we may represent F (A) as an infinite intersection of closed half-planes (see Theorem 1.5.12 of [3]), namely, i θ F (A) = in(e A). (1) H θ∈[0,2π] In particular, we can estimate F (A) simply by drawing the tangent lines θ , L j j = 1, 2,...,k, for a partition 0 = θ < θ < < θk− < θk = 2π of the interval 1 1 · · · 1 [0, 2π]. This is illustrated in our first example below. Example 2.1. Consider the complex Toeplitz matrix 110 i 211 0 A = . 321 1 432 1 The boundary of the numerical range F (A) is sketched in the left part of Figure 1. In the right part of the figure, F (A) is illustrated as an envelope of 120 tangent lines. Here, and in all figures of the paper, the eigenvalues are marked with +’s. In the sequel, our goal is to replace the tangent lines by cubic curves, introducing an envelope-type set that contains the spectrum and lies in the numerical range. We note in passing that another subset of F (A) that contains the eigenvalues, known as the block numerical range, has been studied extensively; see [9] and [10]. There is no tractable relation known to us between the block numerical range and the envelope. 3 Figure 1: The numerical range of a 4 4 Toeplitz matrix. × 3 A cubic curve that bounds the spectrum Let A be an n n complex matrix, and recall that by y Cn we denote a unit × 1 ∈ eigenvector of the hermitian matrix H(A) corresponding to its largest eigenvalue δ1(A). Define also the quantities 2 ∗ v(A) = S(A)y1 2 and u(A) = Im(y1S(A)y1) S(A)y1 2 = v(A). ≤ Adam and Tsatsomeros [1, Theorem 3.1], extending the methodology of [6], ob- tained the following theorem. Theorem 3.1. Let A Cn×n. Then for every eigenvalue λ σ(A), ∈ ∈ (Reλ δ (A))(Imλ u(A))2 (2) − 2 − ≤ (δ (A) Reλ)[v(A) u(A)2 + (Reλ δ (A))(Reλ δ (A))]. 1 − − − 2 − 1 Motivated by the above result, the authors of [1] also introduced and studied the algebraic curve Γ(A) = s + i t : s,t R, (δ (A) s)[(δ (A) s)2 + (u(A) t)2] ∈ 2 − 1 − − + (δ (A) s)(v(A) u(A)2) = 0 1 − − = δ (A)+iu(A) { 1 } ∪ (δ (A) s)(v(A) u(A)2) s + i t = δ (A)+iu(A) : s,t R, δ (A) s + 1 − − = 0 . 1 ∈ 2 − (δ (A) s)2 + (u(A) t)2 1 − − This is a cubic algebraic curve in s,t R, which separates the complex plane into the ∈ regions 2 2 Γin(A) = s + i t : s,t R, (δ (A) s)[(δ (A) s) + (u(A) t) ] ∈ 2 − 1 − − + (δ (A) s)(v(A) u(A)2) 0 . 1 − − ≥ 4 and 2 2 Γout(A) = s + i t : s,t R, (δ (A) s)[(δ (A) s) + (u(A) t) ] ∈ 2 − 1 − − + (δ (A) s)(v(A) u(A)2) < 0 . 1 − − These types of cubic curves have been extensively studied; a suggested general refer- ence is [7]. Below we analyze some of the geometric features of Γ(A). By Theorem 3.1, it is apparent that σ(A) Γin(A). Furthermore, if s > δ (A) or ⊂ 1 s < δ (A), then s + i t (t R) cannot satisfy the defining equation of Γ(A).
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