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 Orthogonality . . 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-Nullspace 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
7.1 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 Jordan Form . . . .............. 587
7.9 Functions of Nondiagonalizable Matrices ..... 599
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