Contents of 
Matrix Analysis and Applied Linear Algebra
Carl D. Meyer
SIAM Philadelphia 2000

Preface ix

I. Linear Equations. . . . . . . . . . . . . . I
1.1 Introduction I
1.2 Gaussian Elimination and Matrices. . . . . . . . 3
1.3 Gauss-Jordan Method. . . . . . . . . . . . . . 15
1.4 Two-Point Boundary Value Problems. . . . . . . 18
1.5 Making Gaussian Elimination Work. . . . . . . . 21
1.6 III-Conditioned Systems. . . . . . . . . . . . . 33

2. Rectangular Systems and Eclielon Forms. .. 41
2.1 Row Echelon Form and Rank. . . . . . . . . . . 41
2.2 Reduced Row Echelon Form. . . . . . . . . . . 47
2.3 Consistency of Linear Systems. . . . . . . . . . 53
2.4 Homogeneous Systems. . . . . . . . . . . . . . 57
2.5 Nonhomogeneous Systems. . . . . . . . . . . .64
2.6 Electrical Circuits. . . . . . . . . . . . . . . . 73

3. Matrix Algebra. . . . . . . . . . . . .. 79
3.1 From Ancient China to Arthur Cayley. . . . . . . 79
3.2 Addition and Transposition. . . . . . . . . . . 81
3.3 Linearity 89
3.4 WhyDoItThisWay 93
3.5 Matrix Multiplication. . . . . . . . . . . . . . 95
3.6 Properties of Matrix Multiplication. . . . . .. 105
3.7 Matrix Inversion. . . . . . . . . . . . . .. 115
3.8 Inverses of Sums and Sensitivity. . . . . . .. 124
3.9 Elementary Matrices and Equivalence. . . . .. 131
3.10 The LV Factorization. . . . . . . . . . . . . 141

4. Vector Spaces. . . . . . . . . . . . . . . 159
4.1 Spaces and Subspaces . . . . . . . . . . . . . 159
4.2 Four Fundamental Subspaces . . . . . . . . .. 169
4.3 Linear Independence. . . . . . . . . . . . . 181
4.4 Basis and Dimension. . . . . . . . . . . .. 194
vi Contents
4.5 More about Rank. . . . . . . . . . . . . . . 210
4.6 Classical Least Squares. . . . . . . . . . . . 223
4.7 Linear Transformations. . . . . . . . . . . . 238
4.8 Change of Basis and Similarity. . . . . . . . . 251
4.9 Invariant Subspaces . . . . . . . . . . . . . . 259

5. Norms, Inner Products, and Ortliogonality . . 269
5.1 Vector Norms 269
5.2 Matrix Norms. . . . . . . . . . . . . . . . 279
5.3 Inner-Product Spaces. . . . . . . . . . . . . 286
5.4 Orthogonal Vectors. . . . . . . . . . . . . . 294
5.5 Gram-Schmidt Procedure. . . . . . . . . . . 307
5.6 Unitary and Orthogonal Matrices. . . . . . . . 320
5.7 Orthogonal Reduction. . . . . . . . . . . . . 341
5.8 Discrete Fourier Transform. . . . . . . . . . . 356
5.9 Complementary Subspaces . . . . . . . . . . . 383
5.10 Range-NuIlspace Decomposition. . . . . . . . 394
5.11 Orthogonal Decomposition. . . . . . . . . . . 403
5.12 Singular Value Decomposition. . . . . . . . . 411
5.13 Orthogonal Projection. . . . . . . . . . . . . 429
5.14 Why Least Squares? . . . . . . . . . . . . . 446
5.15 Angles between Subspaces . . . . . . . . . . . 450

6. Determinants . . 459
6.1 Determinants 459
6.2 Additional Properties of Determinants. . . . . . 475

7. Eigenvalues and Eigenvectors . . . . . . . . 489
7. I Elementary Properties of Eigensystems . . . . . 489
7.2 Diagonalization by Similarity Transformations. . 505
7.3 Functions of Diagonalizable Matrices. . . . . . 525
7.4 Systems of Differential Equations. . . . . . . . 541
7.5 Normal Matrices. . . . . . . . . . . . . . . 547
7.6 Positive Definite Matrices. . . . . . . . . . . 558
7.7 Nilpotent Matrices and Jordan Structure. . . . 574
7.8 JordanForm 587
7.9 Functions of Nondiagonalizable Matrices. . . .. 599
Contents vii
7.10 Difference Equations, Limits, and Summability . . 616
7.11 Minimum Polynomials and Krylov Methods. . . 642

8. Perron-FrobeniusTlieory 661
8.1 Introduction 661
8.2 Positive Matrices. . . . . . . . . . . . . . . 663
8.3 Nonnegative Matrices. . . . . . . . . . . . . 670
8.4 Stochastic Matrices and Markov Chains. . . . . 687

Index 705