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Eigenvalues in Riemannian Geometry
The basic goals of the book are: (i) to introduce the subject to those interested in discovering it, (ii) to coherently present a number of basic techniques and results, currently used in the subject, to those working in it, and (iii) to present some of the results that are attractive in their own right, and which lend themselves to a presentation not overburdened with technical machinery.
Eigenvalues of Matrices
A comprehensive and accessible guide to the calculation of eigenvalues of matrices, ideal for undergraduates, or researchers/engineers in industry.
Eigenvalue and Eigenvector Problems in Applied Mechanics
This book presents, in a uniform way, several problems in applied mechanics, which are analysed using the matrix theory and the properties of eigenvalues and eigenvectors. It reveals that various problems and studies in mechanical engineering produce certain patterns that can be treated in a similar way. Accordingly, the same mathematical apparatus allows us to study not only mathematical structures such as quadratic forms, but also mechanics problems such as multibody rigid mechanics, continuum mechanics, vibrations, elastic and dynamic stability, and dynamic systems. In addition, the book explores a wealth of engineering applications.
The Matrix Eigenvalue Problem
The first in-depth, complete, and unified theoretical discussion of the two most important classes of algorithms for solving matrix eigenvalue problems: QR-like algorithms for dense problems and Krylov subspace methods for sparse problems. The author discusses the theory of the generic GR algorithm, including special cases (for example, QR, SR, HR), and the development of Krylov subspace methods. This book also addresses a generic Krylov process and the Arnoldi and various Lanczos algorithms, which are obtained as special cases. Theoretical and computational exercises guide students, step by step, to the results. Downloadable MATLAB programs, compiled by the author, are available on a supplementary Web site. Readers of this book are expected to be familiar with the basic ideas of linear algebra and to have had some experience with matrix computations. Ideal for graduate students, or as a reference book for researchers and users of eigenvalue codes.
Eigenvalues, Multiplicities and Graphs
The arrangement of nonzero entries of a matrix, described by the graph of the matrix, limits the possible geometric multiplicities of the eigenvalues, which are far more limited by this information than algebraic multiplicities or the numerical values of the eigenvalues. This book gives a unified development of how the graph of a symmetric matrix influences the possible multiplicities of its eigenvalues. While the theory is richest in cases where the graph is a tree, work on eigenvalues, multiplicities and graphs has provided the opportunity to identify which ideas have analogs for non-trees, and those for which trees are essential. It gathers and organizes the fundamental ideas to allow students and researchers to easily access and investigate the many interesting questions in the subject.
Templates for the Solution of Algebraic Eigenvalue Problems
Mathematics of Computing -- Numerical Analysis.
Lanczos Algorithms for Large Symmetric Eigenvalue Computations
First published in 1985, this book presents background material, descriptions, and supporting theory relating to practical numerical algorithms for the solution of huge eigenvalue problems. This book deals with 'symmetric' problems. However, in this book, 'symmetric' also encompasses numerical procedures for computing singular values and vectors of real rectangular matrices and numerical procedures for computing eigenelements of nondefective complex symmetric matrices. Although preserving orthogonality has been the golden rule in linear algebra, most of the algorithms in this book conform to that rule only locally, resulting in markedly reduced memory requirements. Additionally, most of the algorithms discussed separate the eigenvalue (singular value) computations from the corresponding eigenvector (singular vector) computations. This separation prevents losses in accuracy that can occur in methods which, in order to be able to compute further into the spectrum, use successive implicit deflation by computed eigenvector or singular vector approximations.
Eigenvalue Distribution of Large Random Matrices
Random matrix theory is a wide and growing field with a variety of concepts, results, and techniques and a vast range of applications in mathematics and the related sciences. The book, written by well-known experts, offers beginners a fairly balanced collection of basic facts and methods (Part 1 on classical ensembles) and presents experts with an exposition of recent advances in the subject (Parts 2 and 3 on invariant ensembles and ensembles with independent entries). The text includes many of the authors' results and methods on several main aspects of the theory, thus allowing them to present a unique and personal perspective on the subject and to cover many topics using a unified approach...
Operator Theory in Krein Spaces and Nonlinear Eigenvalue Problems
This volume contains a collection of recent original research papers in operator theory in Krein spaces, on generalized Nevanlinna functions, which are closely connected with this theory, and on nonlinear eigenvalue problems. Key topics include: spectral theory for normal operators; perturbation theory for self-adjoint operators in Krein spaces; and, models for generalized Nevanlinna functions.
