## Eigenvalues of matricesEigenvalues of Matrices Francoise Chatelin, Laboratoire Central de Recherches, Thomson-CSF, Orsay, France With exercises by Francoise Cnatelin and Mario Ahues, Universite de Saint-Etienne, France Translated with additional material by Walter Ledermann, University of Sussex The calculation of eigenvalues of matrices is a problem of great practical and theoretical importance with many different types of application. This book provides a modern and complete guide to this subject, at an elementary level, by presenting in matrix notation the fundamental aspects of the theory of linear operators in finite dimensions. This volume is a combination of two books; translations of Professor Chatelin's original and the corresponding book of exercises by Professor Ahues. The exercises are an indispensable complement to the main text. Solutions are furnished for some of the exercises. The book will be of particular value to undergraduate students following courses on numerical analysis and for researchers and practitioners with an interest in this area. |

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### Contents

Elements of Spectral Theory | 61 |

Why Compute Eigenvalues? | 111 |

Error Analysis | 149 |

Copyright | |

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adjoint bases algebraic multiplicity approximation bounds calculated canonical angles Chapter Chatelin Chebyshev iteration Chebyshev polynomials coefficients columns complex compute cond2 condition number Consider convergence Corollary corresponding deduce defined denoted diagonal matrix diagonalisable distinct eigenvalues dominant eigenvalue eigenelements eigenvalue problem ellipse Example Exercise exists function given Hence Hermitian matrix Hessenberg matrix inequality invariant subspace inverse iteration invertible matrix Jordan basis Jordan form Krylov Lanczos method Lemma linear linearly independent matrix of order Newton's method non-zero norm normal obtain orthogonal projection orthonormal basis perturbation positive definite power method precision Proof Let Proposition Prove Q*AQ QR algorithm Rayleigh quotient reader reduced resolvent respectively Schur form Section semi-simple sequence Show simple eigenvalue singular values solution sp(A sp(B spectral projection spectrum subspace iteration sufficiently small Suppose symmetric Theorem Tk(z tridiagonal u*Au unitary matrix upper triangular matrix vector verify whence zero