Matrices and Linear Transformations

Chapter I Matrices and Linear Systems 1.1 Introduction 1.2 Fields and number systems 1.3 Matrices 1.4 Matrix addition and scalar multiplication 1.5 Transposition 1.6 Partitioned matrices 1.7 Special kinds of matrices 1.8 Row equivalence 1.9 Elementary matrices and matrix Inverses 1.10 Column equivalence 1.11 Equivalence Chapter 2 Vector Spaces 2.1 Introduction 2.2 Subspaces 2.3 Linear independence and bases 2.4 The rank of a matrix 2.5 Coordinates and isomorphisms 2.6 Uniqueness theorem for row equivalence. Chapter 3 Determinants 3.1 Definition of the determinant 3.2 The Laplace expansion 3.3 Adjoints and inverses 3.4 Determinants and rank Chapter 4 Linear Transformations 4.1 Definition and examples 4.2 Matrix representation 4.3 Products and inverses 4.4 Change of basis and similarity 4.5 Characteristic vectors and characteristic values 4.6 Orthogonality and length 4.7 Gram-Schmidt process 4.8 Schur's theorem and normal matrices Chapter 5 Similarity: Part I 5.1 The Cayley-Hamilton theorem 5.2 Direct sums and invariant subspaces 5.3 Nilpotent linear operators 5.4 The Jordan canonical form 5.5 Jordan form-continued 5.6 Commutativity (the equation AX = XB) Chapter 6 Polynomials and Polynomial Matrices 6.1 Introduction and review 6.2 Divisibility and irreducibility 6.3 Lagrange interpolation 6.4 Matrices with polynomial elements 6.5 Equivalence over F[x] . 6.6 Equivalence and similarity Chapter 7 Similarity: Part II 7.1 Nonderogatory matrices 7.2 Elementary divisors 7.3 The classical canonical form 7.4 Spectral decomposition 7.5 Polar decomposition Chapter 8 Matrix Analysis 8.1 Sequences and series 8.2 Primary functions 8.3 Matrices of functions 8.4 Systems of linear differential equations Chapter 9 Numerical Methods 9.1 Introduction 9.2 Exact methods for solving AX = K 9.3 Iterative methods for solving AX = K 9.4 Characteristic values and vectors Answers to Selected Exercises Appendix Glossary of Mathematical Symbols Index