Catelog-2
7 The Singular Value Decomposition ()
7.1 Singular Values and Singular Vectors . . . . . . . . . . . . . . 286
7.2 Image Processing by Linear Algebra . . . . . . . . . . . . . . . 287
7.3 Principal Component Analysis () by the . . . . . . . . . 297
8 Linear Transformations
8.1 The Idea of a Linear Transformation . . . . . . . . . . . . . . 302
8.2 The Matrix of a Linear Transformation . . . . . . . . . . . . . . 309
8.3 The Search for a Good Basis . . . . . . . . . . . . . . . . . . 327
9 Linear Algebra in Optimization
9.1 Minimizing a Multivariable Function . . . . . . . . . . . . . . 336
9.2 Backpropagation and Stochastic Gradient Descent . . . . . . . . 346
9.3 Constraints, Lagrange Multipliers, Minimum Norms . . . . . . . 355
9.4 Linear Programming, Game Theory, and Duality . . . . . . . . . 364
10 Learning from Data
10.1 Piecewise Linear Learning Functions . . . . . . . . . . . . . . 370
10.2 Creating and Experimenting . . . . . . . . . . . . . . . . . . 372
10.3 Mean, Variance, and Covariance . . . . . . . . . . . . . . . . 381
Appendix 1 The Ranks of and . . . . . . . . . . . . . . . . 400
Appendix 2 Matrix Factorizations . . . . . . . . . . . . . . . . . . . 401
Appendix 3 Counting Parameters in the Basic Factorizations . . . . . . 403
Appendix 4 Codes and Algorithms for Numerical Linear Algebra . . . . . 404
Appendix 5 The Jordan Form of a Square Matrix . . . . . . . . . . . . 405
Appendix 6 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . 406
Appendix 7 The Condition Number of a Matrix Problem . . . . . . . . . 407
Appendix 8 Markov Matrices and Perron-Frobenius . . . . . . . . . . . 408
Appendix 9 Elimination and Factorization . . . . . . . . . . . . . . . . 410
Appendix 10 Computer Graphics . . . . . . . . . . . . . . . . . . . . 414
Index of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 419
Index of Notations . . . . . . . . . . . . . . . . . . . . . . . . . . 422
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423