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.
Harry Dym, Weizmann Institute of Science, Rehovot, Israel
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.
