Matrix Theory: From Generalized Inverses to Jordan Form

Cover......Page 1
Title page\0......Page 6
Copyright Page......Page 7
Dedication......Page 8
Preface\0......Page 10
Introduction\0......Page 12
Contents\0......Page 16
1.1  Solving Systems of Linear Equations\0......Page 22
1.1.1.1  Floating Point Arithmetic\0......Page 31
1.1.1.2  Arithmetic Operations\0......Page 32
1.1.1.3  Loss of Significance\0......Page 33
1.1.2.1  Creating Matrices in MATLAB\0......Page 34
1.2  The Special Case of \"Square\" Systems\0......Page 38
1.2.1  The Henderson Searle Formulas\0......Page 42
1.2.2  Schur Complements and the Sherman-Morrison-Woodbury Formula\0......Page 45
1.2.3.1  Computing Inverse Matrices\0......Page 58
1.2.4.2  Operation Counts\0......Page 60
2.1  A Brief Review of Gauss Elimination with Back Substitution\0......Page 62
2.1.1.1  Solving Systems of Linear Equations\0......Page 68
2.2  Elementary Matrices\0......Page 70
2.2.1  The Minimal Polynomial\0......Page 78
2.3  The LU and LDU Factorization\0......Page 84
2.3.1.1  The LU Factorization\0......Page 96
2.4  The Ad jugate of a Matrix\0......Page 97
2.5  The Frame Algorithm and the Cayley-Hamilton Theorem\0......Page 102
2.5.1  Digression on Newton\'s Identities\0......Page 106
2.5.2  The Characteristic Polynomial and the Minimal Polynomial\0......Page 111
2.5.4.1  Polynomials in MATLAB\0......Page 116
 5  Generalized Inverses I\0......Page 120
3.1.1.1  The Fundamental Subspaces\0......Page 130
3.2  A Deeper Look at Rank\0......Page 132
3.3  Direct Sums and Idempotents\0......Page 138
3.4  The Index of a Square Matrix\0......Page 149
3.4.1.1  The Standard Nilpotent Matrix\0......Page 168
3.5  Left and Right Inverses\0......Page 169
4.1  Row Reduced Echelon Form and Matrix Equivalence\0......Page 176
4.1.1  Matrix Equivalence\0......Page 181
4.1.2.1  Row Reduced Echelon Form\0......Page 188
4.1.3.1  Pivoting Strategies\0......Page 190
4.1.3.2  Operation Counts\0......Page 191
4.2  The Hermite Echelon Form\0......Page 192
4.3  Full Rank Factorization\0......Page 197
4.4  The Moore-Penrose Inverse\0......Page 200
4.5  Solving Systems of Linear Equations\0......Page 211
4.6  Schur Complements Again (optional)\0......Page 215
5.1  The {1}-Inverse\0......Page 220
5.2  {1,2}-Inverses\0......Page 229
5.3  Constructing Other Generalized Inverses\0......Page 231
5.4  {2}-Inverses\0......Page 238
5.5  The Drazin Inverse\0......Page 244
5.6  The Group Inverse\0......Page 251
6.1  The Normed Linear Space C^n\0......Page 254
6.2  Matrix Norms\0......Page 265
6.2.1.1  Norms\0......Page 273
7.1  The Inner Product Space C^n\0......Page 278
7.2  Orthogonal Sets of Vectors in C^n\0......Page 283
7.3  QR Factorization\0......Page 290
7.3.1 Kung\'s Algorithm\0......Page 295
7.3.2.1  The QR Factorization\0......Page 297
7.4  A Fundamental Theorem of Linear Algebra\0......Page 299
7.5  Minimum Norm Solutions\0......Page 303
7.6  Least Squares\0......Page 306
8.1  Orthogonal Projections\0......Page 312
8.2  The Geometry of Subs paces and the Algebra of Projections\0......Page 320
8.3  The Fundamental Projections of a Matrix\0......Page 330
8.4  Full Rank Factorizations of Projections\0......Page 334
8.5  Affine Projections\0......Page 336
8.6  Quotient Spaces (optional)\0......Page 345
9.1  Eigenstuff\0......Page 350
9.1.1.1  Eigenvalues and Eigenvectors in MATLAB\0......Page 358
9.2  The Spectral Theorem\0......Page 359
9.3  The Square Root and Polar Decomposition Theorems\0......Page 368
10.1  Diagonalization with Respect to Equivalence\0......Page 372
10.2  Diagonali.lation with Respect to Similarity\0......Page 378
10.3  Diagonahzation with Respect to a Unitary\0......Page 392
10.3.1.1  Schur Triangularization\0......Page 397
10.4  The Singular Value Decomposition\0......Page 398
10.4.1.1  The Singular Value Decomposition\0......Page 406
11.1.1  Jordan Blocks\0......Page 410
11.1.2  Jordan Segments\0......Page 413
11.1.2.1  MATLAB Moment\0......Page 416
11.1.3  Jordan Matrices\0......Page 417
11.1.3.1  MATLAB Moment\0......Page 418
11.1.4  Jordan\'s Theorem\0......Page 419
11.1.4.1  Generalized Eigenvectors\0......Page 423
11.2  The Smith Normal Form (optional)\0......Page 443
12.1  Bilinear Forms\0......Page 452
12.2  Matrices Associated to Bilinear Forms\0......Page 458
12.3  Orthogonality\0......Page 461
12.4  Symmetric Bilinear Forms\0......Page 463
12.5  Congruence and Symmetric Matrices\0......Page 468
12.6  Skew-Symmetric Bilinear Forms\0......Page 471
12.7  Tensor Products of Matrices\0......Page 473
12.7.1.1  Tensor Product of Matrices\0......Page 477
A.1  What Is a Scalar?\0......Page 480
A.2  The System of Complex Numbers\0......Page 485
A.3.1.4  Commutative Law of Addition\0......Page 487
A.3.1.9  Existence of Inverses\0......Page 488
A.4  Complex Conjugation, Modulus, and Distance\0......Page 489
A.4.2  Basic Facts about Magnitude\0......Page 490
A.4.3  Basic Properties of Distance\0......Page 491
A.5  The Polar Form of Complex Numbers\0......Page 494
A.6  Polynomials over C\0......Page 501
A.7  Postscript\0......Page 503
B.1  Introduction\0......Page 506
B.2  Matrix Addition\0......Page 508
B.3  Scalar Multiplication\0......Page 510
B.4  Matrix Multiplication\0......Page 511
B.5  Transpose\0......Page 516
B.5.1.1  Matrix Manipulations\0......Page 523
B.6  Submatrices\0......Page 524
B.6.1.1  Getting at Pieces of Matrices\0......Page 527
C.1  Motivation\0......Page 530
C.2  Defining Determinants\0......Page 533
C.3.2  The Cauchy-Binet Theorem\0......Page 538
C.3.3  The Laplace Expansion Theorem\0......Page 541
C.4  The Trace of a Square Matrix\0......Page 549
D.1  Spanning\0......Page 552
D.2  Linear Independence\0......Page 554
D.3  Basis and Dimension\0......Page 555
D.4  Change of Basis\0......Page 559
Index\0......Page 564