Matrix Methods: An Introduction

Matrices. Basic Concepts. Operations. Matrix Multiplication. Special Matrices. Submatrices and Partitioning. Vectors. The Geometry of Vectors. Simultaneous Linear Equations: Linear Systems. Solutions by Substitution. Gaussian Elimination. Pivoting Strategies. Linear Independence. Rank. Theory of Solutions. Appendix. The Inverse: Introduction. Calculating Inverses. Simultaneous Equations. Properties of the Inverse. LU Decomposition. Appendix. Determinants. Expansion by Confactors. Properties of Determinants. Pivotal Condensation. Inversion. Cramer's Rule. Appendix. Eigenvalues and Eigenvectors. Definitions. Eigenvalues. Eigenvectors. Properties of Eigenvalues and Eigenvectors. Linearly Independent Eigenvectors. Power Methods. Real Inner Products. Introduction. Orthonormal Vectors. Projections and QR Decompostions. The QR Algorithm. Least*b1Squares. Matrix Calculus. Well-Defined Functions. Cayley-Hamilton Theorem. Polynomials of Matrices--Distinct Eigenvalues. Polynomials of Matrices--General Case. Fuctions of a Matrix. The Function eAt. Complex Eigenvalues. Properties of eA. Derivatives of a Matrix. Appendix. Linear Differential Equations. Fundamental Form. Reduction of an nth Order Equation. Reduction of a System. Solutions of Systems with Constant Coefficients. Solutions of Systems--General Case. Appendix. Jordan Canonical Forms. Similar Matrices. Diagonalizable Matrices. Functions of Matrices--Diagonalizable Matrices. Generalized Eigenvectors. Chains. Canonical Basis. Jordan Canonical Forms. Functions of Matrices--General Case. The Function eAt. Appendix. Special Matrices. Complex Inner Product. Self-Adjoint Matrices. Real Symmetric Matrices. Orthogonal Matrices. Hermitian Matrices. Unitary Matrices. Summary. Positive Definite Matrices. Answers and Hints to Selected Problems. Index.