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 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 .

View Full Text

Details

  • File Type
    pdf
  • Upload Time
    -
  • Content Languages
    English
  • Upload User
    Anonymous/Not logged-in
  • File Pages
    35 Page
  • File Size
    -

Download

Channel Download Status
Express Download Enable

Copyright

We respect the copyrights and intellectual property rights of all users. All uploaded documents are either original works of the uploader or authorized works of the rightful owners.

  • Not to be reproduced or distributed without explicit permission.
  • Not used for commercial purposes outside of approved use cases.
  • Not used to infringe on the rights of the original creators.
  • If you believe any content infringes your copyright, please contact us immediately.

Support

For help with questions, suggestions, or problems, please contact us