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EXPLICIT BOUNDS for the PSEUDOSPECTRA of VARIOUS CLASSES of MATRICES and OPERATORS Contents 1. Introduction 2 2. Pseudospectra 2

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EXPLICIT BOUNDS for the PSEUDOSPECTRA of VARIOUS CLASSES of MATRICES and OPERATORS Contents 1. Introduction 2 2. Pseudospectra 2 EXPLICIT BOUNDS FOR THE PSEUDOSPECTRA OF VARIOUS CLASSES OF MATRICES AND OPERATORS FEIXUE GONG1, OLIVIA MEYERSON2, JEREMY MEZA3, ABIGAIL WARD4 MIHAI STOICIU5 (ADVISOR) SMALL 2014 - MATHEMATICAL PHYSICS GROUP N×N Abstract. We study the "-pseudospectra σ"(A) of square matrices A ∈ C . We give a complete characterization of the "-pseudospectrum of any 2×2 matrix and describe the asymptotic behavior (as " → 0) of σ"(A) for any square matrix A. We also present explicit upper and lower bounds for the "-pseudospectra of bidiagonal and tridiagonal matrices, as well as for finite rank operators. Contents 1. Introduction 2 2. Pseudospectra 2 2.1. Motivation and Definitions 2 2.2. Normal Matrices 6 2.3. Non-normal Diagonalizable Matrices 7 2.4. Non-diagonalizable Matrices 8 3. Pseudospectra of 2 2 Matrices 10 3.1. Non-diagonalizable× 2 2 Matrices 10 3.2. Diagonalizable 2 2 Matrices× 12 4. Asymptotic Union of Disks× Theorem 15 5. Pseudospectra of N N Jordan Block 20 6. Pseudospectra of bidiagonal× matrices 22 6.1. Asymptotic Bounds for Periodic Bidiagonal Matrices 22 6.2. Examples of Bidiagonal Matrices 29 7. Tridiagonal Matrices 32 Date: October 30, 2014. 1;2;5Williams College 3Carnegie Mellon University 4University of Chicago. 1 2 GONG, MEYERSON, MEZA, WARD, STOICIU 8. Finite Rank operators 33 References 35 1. Introduction The pseudospectra of matrices and operators is an important mathematical object that has found applications in various areas of mathematics: linear algebra, func- tional analysis, numerical analysis, and differential equations. An overview of the main results on pseudospectra can be found in [14]. In this paper we investigate a few classes of matrices and operators and provide explicit bounds on their "-pseudospectra. We study 2 2 matrices, bidiagonal and tridiagonal matrices, as well as finite rank operators. We also describe the asymptotic behavior of the "-pseudospectrum σ" A of any× n n matrix A. The paper is organized as follows: in Section 2 we( give) the three× standard equivalent definitions for the pseudospectrum and we present the \classical" results on "- pseudospectra of normal and diagonalizable matrices (the Bauer-Fike theorems). Section 3 contains a detailed analysis of the "-pseudospectrum of 2 2 matrices, including both the non-diagonalizable case (Subsection 3.1) and the diagonalizable case (Subsection 3.2). The asymptotic behavior (as " 0) of the "-pseudospectrum× of any n n matrix is described in Section 4, where we show (in Theorem 4.1) that, for any square matrix A, the "-pseudospectrum converges,→ as " 0 to a union of disks. × → The main result of Section 4 is applied to several classes of matrices: Jordan blocks (Section 5), bidiagonal matrices (Section 6), and tridiagonal matrices (Section 7). In Section 5 we derive explicit lower bounds for σ" J , where J is any Jordan block, while Section 6 is dedicated to the analysis of arbitrary periodic bidiagonal matrices A. We derive explicit formulas (in terms the coefficients( ) of A) for the asymptotic radii of the "-pseudospectrum of A, as " 0. We continue with a brief investigation of tridiagonal matrices in Section 7, while in the last section (Section 8) we consider finite rank operators and show that the→"-pseudospectrum of an operator of rank 1 m is at most as big as C" m , as " 0. → 2. Pseudospectra 2.1. Motivation and Definitions. The concept of the spectrum of a matrix A N×N C provides a fundamental tool for understanding the behavior of A. As is well- known, a complex number z C is in the spectrum of A (denoted σ A ) whenever∈ zI A (which we will denote as z A) is not invertible, i.e., the characteristic polynomial of A has z as a root.∈ As slightly perturbing the coefficients( ) of A will change− the roots of the characteristic− polynomial, the property of \membership in the set of eigenvalues" is not well-suited for many purposes, especially those in numerical analysis. We thus want to find a characterization of when a complex EXPLICIT BOUNDS FOR THE PSEUDOSPECTRA OF MATRICES AND OPERATORS 3 number is close to an eigenvalue, and we do this by considering the set of complex numbers z such that z A −1 is large, where the norm here is the usual operator norm induced by the Euclidean norm, i.e. Y( − ) Y A sup Av : YvY=1 The motivation for considering thisY Y question= Y comesY from the following observation: N×N Proposition 2.1. Let A C , λ an eigenvalue of A, and zn be a sequence of −1 complex numbers converging to λ with zn z for all n. Then zn A as n . ∈ ≠ Y( − ) Y → ∞ Proof.→ ∞ Let v be the eigenvector corresponding to λ with norm 1, so Av λv. Define zn A v vn = zn A v ( − ) (note that zn A v 0 by assumption).= Now, note that Y( − ) Y z A −1 z A −1v ( − ) ≠ n n n z A −1 z A v Y( − ) Y ≥ Y( n− ) Yn zn A v ( − ) ( − ) = ] v ] Y( − ) Y zn A v = ] 1 ] Y( −; ) Y λ zn and since the last quantity gets arbitrarily= large for large n, we obtain the result. S − S We call the operator z A −1 the resolvent of A. The observation that the norm of the resolvent is large when z is close to an eigenvalue of A leads us to the first definition of the "-pseudospectrum( − ) of an operator. N×N Definition 2.1. Let A C , and let " 0. The "-pseudospectrum of A is the set of z C such that ∈ z A −1> 1 " ∈ Note that the boundary of the "Y(-pseudospectrum− ) Y > ~ is exactly the 1 " level curve of the function z z A −1 . Fig. 2.1 depicts the behavior of this function near the eigenvalues, as described in Prop. 2.1. ~ ↦ Y( − ) Y The resolvent norm has singularities in the complex plane, and as we approach these points, the resolvent norm grows to infinity. Conversely, if z A −1 approaches infinity, then z must approach some eigenvalue of A [14, Thm 2.4]. Y( − ) Y (It is also possible to develop a theory of pseudo-spectrum for operators on Banach spaces, and it is important to note that this converse does not necessarily hold for such operators; that is, there are operators [4, 5] such that z A −1 approaches infinity, but z does not approach the spectrum of A.) Y( − ) Y The second definition of the "-pseudospectrum arises from eigenvalue perturbation theory [8]. 4 GONG, MEYERSON, MEZA, WARD, STOICIU Figure 2.1. Contour Plot of Resolvent Norm N×N Definition 2.2. Let A C . The "-pseudospectrum of A is the set of z C such that ∈ z σ A E ∈ for some E with E ". ∈ ( + ) Finally, the thirdY definitionY < of the "-pseudospectum is the following: N×N Definition 2.3. Let A C . The "-pseudospectrum of A is the set of z C such that ∈ z A v " ∈ for some unit vector v. Y( − ) Y < This definition is similar to our first definition in that it quantifies how close z is to an eigenvalue of A. In addition to this, it also gives us the notion of an "-pseudoeigenvector. Theorem 2.1 (Equivalence of the definitions of pseudospectra). For any matrix N×N A C , the three definitions above are equivalent. ∈ The proof of this theorem follows [14, §2]. Proof. Define the following sets −1 −1 M" z C z A " N z z σ A E for some E N×N ; E " " C = { ∈ S Y( − ) Y > C } N P" z C z A v " for some v C ; v 1 = { ∈ S ∈ ( + ) ∈ Y Y < } To show equivalence,= { we∈ willS Y( show− ) thatY <M" N" P"∈ M".Y Y = } N" P" ⊆ ⊆ ⊆ Let⊆z N". Let E be such that z σ A E , with E ". Let v be an eigenvector of norm one for z, that is v 1 and A E v zv. Then ∈ ∈ ( + ) Y Y < z A v z A E v Ev E v " Y Y = ( + ) = and hence z P . " Y( − ) Y = Y( − − ) + Y ≤ Y YY Y < ∈ EXPLICIT BOUNDS FOR THE PSEUDOSPECTRA OF MATRICES AND OPERATORS 5 P" M" N Let⊆z P". Let v C be such that v 1 and z A v ". Observe z A v ∈ ∈ z A v zY YA= v Y( − ) suY < z A v ( − ) (z−(A)−v ) = Y( − ) Y = −1 −1 where we define u Y(z−A)vY and s z A vY(. Then,− ) Y we see that s v z A u, which implies that = = Y( − ) Y = ( − ) z A −1 z A −1u s−1 v s−1 "−1 and hence z M".Y( − ) Y ≥ Y( − ) Y = S SY Y = > M" N" ∈ Let ⊆z M". We will construct an E with E " and z σ A E . Since −1 −1 −1 z A supYvY=1 z A v " , there exists some v with v 1 such that z A∈ −1v "−1. Observe Y Y < ∈ ( + ) Y( − ) Y = Y( − ) Y > Y Y = z A −1v Y( − ) Y > z A −1v z A −1v z A −1v u z A −1v ( − ) ( − ) = Y((z−A−)−1)v Y v = Y( − ) Y where u is the unit vector Y(z−A)−1vY . SoY(Y(z−−A))−1vY Y z A u, from which we have v zu Au = ( − ) z A −1v Define E v u∗. Note Eu= + v and E ". Then we have zu Y(z−A)−1vY Y(zY(−A)−1vY) Y Au Eu, and so z σ A E . = = Y Y < = + ∈ ( + ) As all three definitions are now shown to be equivalent, we can unambiguously denote the "-pseudospectrum of A as σ" A .
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