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Matrix Algorithms, Volume II: Eigensystems

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This book, Eigensystems, is the second volume in a projected five-volume series entitled Matrix Algorithms. The first volume treated basic decompositions. The three following this volume will treat iterative methods for linear systems, sparse direct methods, and special topics, including fast algorithms for structured matrices.

My intended audience is the nonspecialist whose needs cannot be satisfied by black boxes. It seems to me that these people will be chiefly interested in the methods themselves— how they are derived and how they can be adapted to particular problems. Consequently, the focus of the series is on algorithms, with such topics as roundingerror analysis and perturbation theory introduced impromptu as needed. My aim is to bring the reader to the point where he or she can go to the research literature to augment what is in the series.

The series is self-contained. The reader is assumed to have a knowledge of elementary analysis and linear algebra and a reasonable amount of programming experience— about what you would expect from a beginning graduate engineer or an advanced undergraduate in an honors program. Although strictly speaking the individual volumes are not textbooks, they are intended to teach, and my guiding principle has been that if something is worth explaining it is worth explaining fully. This has necessarily restricted the scope of the series, but I hope the selection of topics will give the reader a sound basis for further study.

The subject of this volume is computations involving the eigenvalues and eigenvectors of a matrix. The first chapter is devoted to an exposition of the underlying mathematical theory. The second chapter treats the QR algorithm, which in its various manifestations has become the universal workhorse in this field. The third chapter deals with symmetric matrices and the singular value decomposition. The second and third chapters also treat the generalized eigenvalue problem.
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