## Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and OperatorsPure 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 |

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### Contents

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 |

Toeplitz Matrices | 47 |

7 Toeplitz matrices and boundary pseudomodes | 49 |

32 Stability of the method of lines | 302 |

33 Stiffness of ODEs | 314 |

34 GKSstability of boundary conditions | 322 |

Random Matrices | 331 |

35 Random dense matrices | 333 |

36 HatanoNelson matrices and localization | 339 |

37 Random Fibonacci matrices | 351 |

38 Random triangular matrices | 359 |

8 Twisted Toeplitz matrices and wave packet pseudomodes | 62 |

9 Variations on twisted Toeplitz matrices | 74 |

Differential Operators | 85 |

10 Differential operators and boundary pseudomodes | 87 |

11 Variable coefficients and wave packet pseudomodes | 98 |

12 Advectiondiffusion operators | 115 |

13 LewyHörmander nonexistence of solutions | 126 |

Transient Effects and Nonnormal Dynamics | 133 |

14 Overview of 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 |

Fluid Mechanics | 193 |

20 Stability of fluid flows | 195 |

21 A model of transition to turbulence | 207 |

22 OrrSommerfeld and Airy operators | 215 |

23 Further problems in fluid mechanics | 224 |

Matrix Iterations | 229 |

24 GaussSeidel 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 |

Numerical Solution of Differential Equations | 287 |

30 Spectral differentiation matrices | 289 |

31 Nonmodal instability of PDE discretizations | 295 |

Computation of Pseudospectra | 369 |

39 Computation of matrix pseudospectra | 371 |

40 Projection for largescale matrices | 381 |

41 Other computational techniques | 391 |

42 Pseudospectral abscissae and radii | 397 |

43 Discretization of continuous operators | 405 |

44 A flow chart of pseudospectra algorithms | 416 |

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 |

Further Examples and Applications | 499 |

55 Companion matrices and zeros of polynomials | 501 |

56 Markov chains and the cutoff phenomenon | 508 |

57 Card shuffling | 519 |

58 Population ecology | 526 |

59 The PapkovichFadle operator | 534 |

60 Lasers | 542 |

555 | |

597 | |