Matrix Analysis and Applied Linear Algebra

Table of Contents......Page 2
Preface......Page 6
Introduction......Page 10
Gaussian Elimination and Matrices......Page 12
Gauss-Jordan Method......Page 24
Two-Point Boundary Value Problems......Page 27
Making Gaussian Elimination Work......Page 30
Ill-Conditioned Systems......Page 42
Row Echelon Form and Rank......Page 50
Reduced Row Echelon Form......Page 56
Consistency of Linear Systems......Page 62
Homogeneous Systems......Page 66
Nonhomogeneous Systems......Page 73
Electrical Circuits......Page 82
From Ancient China to Arthur Cayley......Page 88
Addition and Transposition......Page 90
Linearity......Page 98
Why Do It This Way......Page 102
Matrix Multiplication......Page 104
Properties of Matrix Multiplication......Page 114
Matrix Inversion......Page 124
Inverses of Sums and Sensitivity......Page 133
Elementary Matrices and Equivalence......Page 140
The LU Factorization......Page 150
Spaces and Subspaces......Page 168
Four Fundamental Subspaces......Page 178
Linear Independence......Page 190
Basis and Dimension......Page 203
More About Rank......Page 219
Classical Least Squares......Page 232
Linear Transformations......Page 247
Change of Basis and Similarity......Page 260
Invariant Subspaces......Page 268
Vector Norms......Page 278
Matrix Norms......Page 288
Inner-Product Spaces......Page 295
Orthogonal Vectors......Page 303
Gram-Schmidt Procedure......Page 316
Unitary and Orthogonal Matrices......Page 329
Orthogonal Reduction......Page 350
Discrete Fourier Transform......Page 365
Complementary Subspaces......Page 392
Range-Nullspace Decomposition......Page 403
Orthogonal Decomposition......Page 412
Singular Value Decomposition......Page 420
Orthogonal Projection......Page 438
Why Least Squares?......Page 455
Angles Between Subspaces......Page 459
Determinants......Page 468
Additional Properties of Determinants......Page 484
Elementary Properties of Eigensystems......Page 498
Diagonalization by Similarity Transformations......Page 514
Functions of Diagonalizable Matrices......Page 534
Systems of Differential Equations......Page 550
Normal Matrices......Page 556
Positive Definite Matrices......Page 567
Nilpotent Matrices and Jordan Structure......Page 583
Jordan Form......Page 596
Functions of Nondiagonalizable Matrices......Page 608
Difference Equations, Limits, and Summability......Page 625
Minimum Polynomials and Krylov Methods......Page 651
Introduction......Page 670
Positive Matrices......Page 672
Nonnegative Matrices......Page 679
Stochastic Matrices and Markov Chains......Page 696
Index......Page 714