Pure and applied mathematicians, physicists, scientists, and engineers use matrices and operators and their eigenvalues in quantum mechanics, fluid mechanics, structural analysis, acoustics, ecology, numerical analysis, and many other areas. However, in some applications the usual analysis based on eigenvalues fails. For example, eigenvalues are often ineffective for analyzing dynamical systems such as fluid flow, Markov chains, ecological models, and matrix iterations. That's where this book comes in. This is the authoritative work on nonnormal matrices and operators, written by the authorities who made them famous. Each of the sixty sections is written as a self-contained essay. Each document is a lavishly illustrated introductory survey of its topic, complete with beautiful numerical experiments and all the right references. The breadth of included topics and the numerous applications that provide links between fields will make this an essential reference in mathematics and related sciences.
Lloyd N. Trefethen is Professor of Numerical Analysis and Head of the Numerical Analysis Group at the University of Oxford. Mark Embree is Assistant Professor of Computational and Applied Mathematics at Rice University.
Preface xiii Acknowledgments xv I. Introduction 1 1.Eigenvalues 3 2.Pseudospectra of matrices 12 3.A matrix example 22 4.Pseudospectra of linear operators 27 5.An operator example 34 6.History of pseudospectra 41 II. Toeplitz Matrices 47 7.Toeplitz matrices and boundary pseudomodes 49 8.Twisted Toeplitz matrices and wave packet pseudomodes 62 9.Variations on twisted Toeplitz matrices 74 III. Differential Operators 85 10.Differential operators and boundary pseudomodes 87 11.Variable coeffcients and wave packet pseudomodes 98 12.Advection-diffusion operators 115 13.Lewy Hormander nonexistence of solutions 126 IV. Transient Effects and Nonnormal Dynamics 133 14.Overviewof transients and pseudospectra 135 15.Exponentials of matrices and operators 148 16.Powers of matrices and operators 158 17.Numerical range, abscissa, and radius 166 18.The Kreiss Matrix Theorem 176 19.Growth bound theorem for semigroups 185 V. Fluid Mechanics 193 20.Stability of fluid flows 195 21.A model of transition to turbulence 207 22.Orr--Sommerfeld and Airy operators 215 23.Further problems in fluid mechanics 224 VI. Matrix Iterations 229 24.Gauss--Seidel and SOR iterations 231 25.Upwind effects and SOR convergence 237 26.Krylov subspace iterations 244 27.Hybrid iterations 254 28.Arnoldi and related eigenvalue iterations 263 29.The Chebyshev polynomials of a matrix 278 VII. Numerical Solution of Differential Equations 287 30.Spectral differentiation matrices 289 31.Nonmodal instability of PDE discretizations 295 32.Stability of the method of lines 302 33.Stiffness of ODEs 314 34.GKS-stability of boundary conditions 322 VIII. Random Matrices 331 35.Random dense matrices 333 36.Hatano--Nelson matrices and localization 339 37.Random Fibonacci matrices 351 38.Random triangular matrices 359 IX. Computation of Pseudospectra 369 39.Computation of matrix pseudospectra 371 40.Projection for large-scale matrices 381 41.Other computational techniques 391 42.Pseudospectral abscissae and radii 397 43.Discretization of continuous operators 405 44.A flowchart of pseudospectra algorithms 416 X. Further Mathematical Issues 421 45.Generalized eigenvalue problems 423 46.Pseudospectra of rectangular matrices 430 47.Do pseudospectra determine behavior? 437 48.Scalar measures of nonnormality 442 49.Distance to singularity and instability 447 50.Structured pseudospectra 458 51.Similarity transformations and canonical forms 466 52.Eigenvalue perturbation theory 473 53.Backward error analysis 485 54.Group velocity and pseudospectra 492 XI. Further Examples and Applications 499 55.Companion matrices and zeros of polynomials 501 56.Markov chains and the cutoff phenomenon 508 57.Card shuffing 519 58.Population ecology 526 59.The Papkovich--Fadle operator 534 60.Lasers 542 References 555 Index 597