作者：Ilse C. F. Ipsen
North Carolina State University
Raleigh, North Carolina

Contents
Preface ix
Introduction xiii
1 Matrices 1
1.1 What Is a Matrix? . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Scalar Matrix Multiplication . . . . . . . . . . . . . . . . . . 4
1.3 Matrix Addition . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Inner Product (Dot Product) . . . . . . . . . . . . . . . . . . 5
1.5 Matrix Vector Multiplication . . . . . . . . . . . . . . . . . 6
1.6 Outer Product . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.7 Matrix Multiplication . . . . . . . . . . . . . . . . . . . . . 9
1.8 Transpose and Conjugate Transpose . . . . . . . . . . . . . . 12
1.9 Inner and Outer Products, Again . . . . . . . . . . . . . . . 14
1.10 Symmetric and Hermitian Matrices . . . . . . . . . . . . . . 15
1.11 Inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.12 Unitary and Orthogonal Matrices . . . . . . . . . . . . . . . 19
1.13 Triangular Matrices . . . . . . . . . . . . . . . . . . . . . . 20
1.14 Diagonal Matrices . . . . . . . . . . . . . . . . . . . . . . . 22
2 Sensitivity, Errors, and Norms 23
2.1 Sensitivity and Conditioning . . . . . . . . . . . . . . . . . 23
2.2 Absolute and Relative Errors . . . . . . . . . . . . . . . . . 25
2.3 Floating Point Arithmetic . . . . . . . . . . . . . . . . . . . 26
2.4 Conditioning of Subtraction . . . . . . . . . . . . . . . . . . 27
2.5 Vector Norms . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.6 Matrix Norms . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.7 Conditioning of Matrix Addition and Multiplication . . . . . 37
2.8 Conditioning of Matrix Inversion . . . . . . . . . . . . . . . 38
3 Linear Systems 43
3.1 The Meaning of Ax = b . . . . . . . . . . . . . . . . . . . . 43
3.2 Conditioning of Linear Systems . . . . . . . . . . . . . . . . 44
3.3 Solution of Triangular Systems . . . . . . . . . . . . . . . . 51
3.4 Stability of Direct Methods . . . . . . . . . . . . . . . . . . 52
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3.5 LU Factorization . . . . . . . . . . . . . . . . . . . . . . . . 58
3.6 Cholesky Factorization . . . . . . . . . . . . . . . . . . . . 63
3.7 QR Factorization . . . . . . . . . . . . . . . . . . . . . . . . 68
3.8 QR Factorization of Tall and Skinny Matrices . . . . . . . . 73
4 Singular Value Decomposition 77
4.1 Extreme Singular Values . . . . . . . . . . . . . . . . . . . . 79
4.2 Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.3 Singular Vectors . . . . . . . . . . . . . . . . . . . . . . . . 86
5 Least Squares Problems 91
5.1 Solutions of Least Squares Problems . . . . . . . . . . . . . 91
5.2 Conditioning of Least Squares Problems . . . . . . . . . . . 95
5.3 Computation of Full Rank Least Squares Problems . . . . . . 101
6 Subspaces 105
6.1 Spaces of Matrix Products . . . . . . . . . . . . . . . . . . . 106
6.2 Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.3 Intersection and Sum of Subspaces . . . . . . . . . . . . . . 111
6.4 Direct Sums and Orthogonal Subspaces . . . . . . . . . . . . 115
6.5 Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
Index 121