Godunov S.K. Modern Aspects of Linear Algebra

Providence: American Mathematical Society, 1998. — 319 p.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 analysisIntroduction:Euclidean linear spaces
Orthogonal and unitary linear transformations
Orthogonal and unitary transformations.
Singular values
Matrices of operators in the Euclidean space: Unitary similar transformations.
The Schur theorem
Alternation theorems
The Weyl inequalities
Variational principles
Resolvent and dichotomy of spectrum
Quadratic forms in the spectrum dichotomy problem
Matrix equations and projections
The Hausdorff set of a matrix
Application of spectral analysis.
The most important algorithms: Matrix operators as models of differential operators
Application of the theory of functions of complex variable
Computational algorithms of spectral analysis