A Survey of Matrix Theory and Matrix Inequalities

Concise yet comprehensive survey covers broad range of topics: convexity and matrices, localization of characteristic roots, proofs of classical theorems and results in contemporary research literature, much more. Undergraduate-level. 1969 edition. Bibliography.

Written for advanced undergraduate students, this highly regarded book presents an enormous amount of information in a concise and accessible format. Beginning with the assumption that the reader has never seen a matrix before, the authors go on to provide a survey of a substantial part of the field, including many areas of modern research interest.Part One of the book covers not only the standard ideas of matrix theory, but ones, as the authors state, "that reflect our own prejudices," among them Kronecker products, compound and induced matrices, quadratic relations, permanents, incidence matrices and generalizations of commutativity.Part Two begins with a survey of elementary properties of convex sets and polyhedra and presents a proof of the Birkhoff theorem on doubly stochastic matrices. This is followed by a discussion of the properties of convex functions and a list of classical inequalities. This material is then combined to yield many of the interesting matrix inequalities of Weyl, Fan, Kantorovich and others. The treatment is along the lines developed by these authors and their successors and many of their proofs are included. This chapter contains an account of the classical Perron Frobenius-Wielandt theory of indecomposable nonnegative matrices and ends with some important results on stochastic matrices.Part Three is concerned with a variety of results on the localization of the characteristic roots of a matrix in terms of simple functions of its entries or of entries of a related matrix. The presentation is essentially in historical order, and out of the vast number of results in this field the authors have culled those that seemed most interesting or useful. Readers will find many of the proofs of classical theorems and a substantial number of proofs of results in contemporary research literature.

I. SURVEY OF MATRIX THEORY 1. INTRODUCTORY CONCEPTS Matrices and vectors. Matrix operations. Inverse. Matrix and vector operations. Examples. Transpose. Direct sum and block multiplication. Examples. Kronecker product. Example. 2. NUMBERS ASSOCIATED WITH MATRICES Notation. Submatrices. Permutations. Determinants. The quadratic relations among subdeterminants. Examples. Compound matrices. Symmetric functions; trace. Permanents. Example. Properties of permanents. Induced matrices. Characteristic polynomial. Examples. Characteristic roots. Examples. Rank. Linear combinations. Example. Linear dependence; dimension. Example. 3. LINEAR EQUATIONS AND CANONICAL FORMS Introduction and notation. Elementary operations. Example. Elementary matrices. Example. Hermite normal form. Example. Use of the Hermite normal form in solving Ax = b. Example. Elementary column operations and matrices. Examples. Characteristic vectors. Examples. Conventions for polynomial and integral matrices. Determinantal divisors. Examples. Equivalence. Example. Invariant factors. Elementary divisors. Examples. Smith normal form. Example. Similarity. Examples. Elementary divisors and similarity. Example. Minimal polynomial. Companion matrix. Examples. Irreducibility. Similarity to a diagonal matrix. Examples. 4. "SPECIAL CLASSES OF MATRICES, COMMUTATIVITY" Bilinear functional. Examples. Inner product. Example. Orthogonality. Example. Normal matrices. Examples. Circulant. Unitary similarity. Example. Positive definite matrices. Example. Functions of normal matrices. Examples. Exponential of a matrix. Functions of an arbitrary matrix. Example. Representation of a matrix as a function of other matrices. Examples. Simultaneous reduction of commuting matrices. Commutativity. Example. Quasi-commutativity. Example. Property L. Examples. Miscellaneous results on commutativity. 5. CONGRUENCE Definitions. Triple diagonal form. Congruence and elementary operations. Example. Relationship to quadratic forms. Example. Congruence properties. Hermitian congruence. Example. Triangular product representation. Example. Conjunctive reduction of skew-hermitian matrices. Conjunctive reduction of two hermitian matrices.II. CONVEXITY AND MATRICES 1. CONVEX SETS Definitions. Examples. Intersection property. Examples. Convex polyhedrons. Example. Birkhoff theorem. Simplex. Examples. Dimension. Example. Linear functionals. Example. 2. CONVEX FUNCTIONS Definitions. Examples. Properties of convex functions. Examples. 3. CLASSICAL INEQUALITIES Power means. Symmetric functions. Hölder inequality. Minkowski inequality. Other inequalities. Example. 4. CONVEX FUNCTIONS AND MATRIX INEQUALITIES Convex functions of matrices. Inequalities of H. Weyl. Kantorovich inequality. More inequalities. Hadamard product. 5. NONNEGATIVE MATRICES Introduction. Indecomposable matrices. Examples. Fully indecomposable matrices. Perron-Frobenius theorem. Example. Nonnegative matrices. Examples. Primitive matrices. Example. Doubly stochastic matrices. Examples. Stochastic matrices.III. LOCALIZATION OF CHARACTERISTIC ROOTS 1. BOUNDS FOR CHARACTERISTIC ROOTS Introduction. Bendixson's theorems. Hirsch's theorems. Schur's inequality (1909). Browne's theorem. Perron's theorem. Schneider's theorem. 2. REGIONS CONTAINING CHARACTERISTIC ROOTS OF A GENERAL MATRIX. Lévy-Desplanques theorem. Gersgorin discs. &