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Full Description
This book is based largely on courses that the author taught at the Feinberg Graduate School of the Weizmann Institute. It conveys in a user-friendly way the basic and advanced techniques of linear algebra from the point of view of a working analyst. The techniques are illustrated by a wide sample of applications and examples that are chosen to highlight the tools of the trade. In short, this is material that the author has found to be useful in his own research and wishes that he had been exposed to as a graduate student. Roughly the first quarter of the book reviews the contents of a basic course in linear algebra, plus a little. The remaining chapters treat singular value decompositions, convexity, special classes of matrices, projections, assorted algorithms, and a number of applications. The applications are drawn from vector calculus, numerical analysis, control theory, complex analysis, convex optimization, and functional analysis. In particular, fixed point theorems, extremal problems, best approximations, matrix equations, zero location and eigenvalue location problems, matrices with nonnegative entries, and reproducing kernels are discussed. This new edition differs significantly from the second edition in both content and style. It includes a number of topics that did not appear in the earlier edition and excludes some that did. Moreover, most of the material that has been adapted from the earlier edition has been extensively rewritten and reorganized.
Contents
Prerequisites
Dimension and rank
Gaussian elimination
Eigenvalues and eigenvectors
Towards the Jordan decomposition
The Jordan decomposition
Determinants
Companion matrices and circulants
Inequalities
Normed linear spaces
Inner product spaces
Orthogonality
Normal matrices
Projections, volumes, and traces
Singular value decomposition
Positive definite and semidefinite matrices
Determinants redux
Applications
Discrete dynamical systems
Continuous dynamical systems
Vector-valued functions
Fixed point theorems
The implicit function theorem
Extremal problems
Newton's method
Matrices with nonnegative entries
Applications of matrices with nonnegative entries
Eigenvalues of Hermitian matrices
Singular values redux I
Singular values redux II
Approximation by unitary matrices
Linear functionals
A minimal norm problem
Conjugate gradients
Continuity of eigenvalues
Eigenvalue location problems
Matrix equations
A matrix completion problem
Minimal norm completions
The numerical range
Riccati equations
Supplementary topics
Toeplitz, Hankel, and de Branges
Bibliography
Notation index
Subject index.
