Graham A. Nonnegative Matrices and Applicable Topics in Linear Algebra 2019

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Graham A. Nonnegative Matrices and Applicable Topics in Linear Algebra 2019

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Textbook in PDF and DJVU formats Nonnegative matrices is an increasingly important subject in economics, control theory, numerical analysis, Markov chains, and other areas. This concise treatment is directed toward undergraduates who lack specialized knowledge at the postgraduate level of mathematics and related fields, such as mathematical economics and operations research. An Introductory Survey encompasses some aspects of matrix theory and its applications and other relevant topics in linear algebra, including certain facets of graph theory. Subsequent chapters cover various points of the theory of normal matrices, comprising unitary and Hermitian matrices, and the properties of positive definite matrices. An exploration of the main topic, nonnegative matrices, is followed by a discussion of M-matrices. The final chapter examines stochastic, genetic, and economic models. The important concepts are illustrated by simple worked examples. Problems appear at the conclusion of most chapters, with solutions at the end of the book. Preface Examples of Notation Used Introductory Survey Introduction Notation Submatrices and Minors Expressing a Singular Matrix as a Product Determinants The Derivative of a Determinant The Characteristic Equation The Adjoint of the Characteristic Matrix Spectral Decomposition Inner Product and Norms The Trace Function Permutation Matrices and Irreducible Matrices Some Aspects of the Theory of Graphs Matrix Convergence Some Matrix Types Introduction Unitary Matrices Hermitian Matrices Normal Matrices Problems Positive Definite Matrices Introduction Quadratic Forms Reductions to a Canonical Form A Geometrical Application Problems Positive Definite Matrices Nonnegative Matrices Introduction Terminology and Notation The Perron–Frobenius Theorem for a Positive Matrix Irreducible Matrices Cyclic Matrices Reducible Matrices Problems M-Matrices Introduction Non-singular M-Matrices Regular Splitting and Solving Simultaneous Equations Problems Finite Markov Chains and Stochastic Matrices Some Applications of Nonnegative Matrices Introduction Some Genetic Models Some Economic Models Some Markov Chain Models Appendix Solutions to Problems References Index