This book discusses fundamental ideas of linear algebra. The author presents the spectral theory of nonselfadjoint matrix operators and matrix pencils in a finite dimensional Euclidean space. Statements of computational problems and brief descriptions of numerical algorithms, some of them nontraditional, are given.

Proved in detail are classical problems that are not usually found in standard university courses. In particular, the material shows the role of delicate estimates for the resolvent of an operator and underscores the need for the study and use of such estimates in numerical analysis.

• Cover1
• Title page6
• Contents8
• Preface12
• Introduction18
• Euclidean linear spaces20
• Orthogonal and unitary linear transformations32
• Orthogonal and unitary transformations. Singular values42
• Matrices of operators in the Euclidean space62
• Unitary similar transformations. The Schur theorem64
• Alternation theorems76
• The Weyl inequalities104
• Variational principles118
• Resolvent and dichotomy of spectrum140
• Quadratic forms in the spectrum dichotomy problem156
• Matrix equations and projections168
• The Hausdorff set of a matrix200
• Application of spectral analysis. The most important algorithms230
• Matrix operators as models of differential operators232
• Application of the theory of functions of complex variables252
• Computational algorithms of spectral analysis274
• Bibliography318
• Index320
• Back Cover321