In 1990, the National Science Foundation recommended that every college mathematics curriculum should include a second course in linear algebra. In answer to this recommendation, Matrix Theory: From Generalized Inverses to Jordan Form provides the material for a second semester of linear algebra that probes introductory linear algebra concepts while also exploring topics not typically covered in a sophomore-level class.Tailoring the material to advanced undergraduate and beginning graduate students, the authors offer instructors flexibility in choosing topics from the book. The text first focuses on the central problem of linear algebra: solving systems of linear equations. It then discusses LU factorization, derives Sylvester's rank formula, introduces full-rank factorization, and describes generalized inverses. After discussions on norms, QR factorization, and orthogonality, the authors prove the important spectral theorem. They also highlight the primary decomposition theorem, Schur's triangularization theorem, singular value decomposition, and the Jordan canonical form theorem. The book concludes with a chapter on multilinear algebra.With this classroom-tested text students can delve into elementary linear algebra ideas at a deeper level and prepare for further study in matrix theory and abstract algebra.
Author(s): Robert Piziak, Patrick L. Odell
Edition: 1
Publisher: Taylor & Francis Group, LLC
Year: 2007
Language: English
Pages: 565
Tags: Математика;Линейная алгебра и аналитическая геометрия;Линейная алгебра;Матрицы и определители;
Cover page......Page 1
Title page......Page 2
Preface......Page 6
Introduction......Page 8
Contents......Page 12
1.1 Solving Systems of Linear Equations......Page 18
1.1.1.1 Floating Point Arithmetic......Page 27
1.1.1.2 Arithmetic Operations......Page 28
1.1.1.3 Loss of Significance......Page 29
1.1.2.1 Creating Matrices in MATLAB......Page 30
1.2 The Special Case of "Square" Systems......Page 34
1.2.1 The Henderson Searle Formulas......Page 38
1.2.2 Schur Complements and the Sherman-Morrison-Woodbury Formula......Page 41
1.2.3.1 Computing Inverse Matrices......Page 54
1.2.4.2 Operation Counts......Page 56
2.1 A Brief Review of Gauss Elimination with Back Substitution......Page 58
2.1.1.1 Solving Systems of Linear Equations......Page 64
2.2 Elementary Matrices......Page 66
2.2.1 The Minimal Polynomial......Page 74
2.3 The LU and LDU Factorization......Page 80
2.3.1.1 The LU Factorization......Page 92
2.4 The Ad jugate of a Matrix......Page 93
2.5 The Frame Algorithm and the Cayley-Hamilton Theorem......Page 98
2.5.1 Digression on Newton's Identities......Page 102
2.5.2 The Characteristic Polynomial and the Minimal Polynomial......Page 107
2.5.4.1 Polynomials in MATLAB......Page 112
5 Generalized Inverses I......Page 116
3.1.1.1 The Fundamental Subspaces......Page 126
3.2 A Deeper Look at Rank......Page 128
3.3 Direct Sums and Idempotents......Page 134
3.4 The Index of a Square Matrix......Page 145
3.4.1.1 The Standard Nilpotent Matrix......Page 164
3.5 Left and Right Inverses......Page 165
4.1 Row Reduced Echelon Form and Matrix Equivalence......Page 172
4.1.1 Matrix Equivalence......Page 177
4.1.2.1 Row Reduced Echelon Form......Page 184
4.1.3.1 Pivoting Strategies......Page 186
4.1.3.2 Operation Counts......Page 187
4.2 The Hermite Echelon Form......Page 188
4.3 Full Rank Factorization......Page 193
4.4 The Moore-Penrose Inverse......Page 196
4.5 Solving Systems of Linear Equations......Page 207
4.6 Schur Complements Again (optional)......Page 211
5.1 The {1}-Inverse......Page 216
5.2 {1,2}-Inverses......Page 225
5.3 Constructing Other Generalized Inverses......Page 227
5.4 {2}-Inverses......Page 234
5.5 The Drazin Inverse......Page 240
5.6 The Group Inverse......Page 247
6.1 The Normed Linear Space C^n......Page 250
6.2 Matrix Norms......Page 261
6.2.1.1 Norms......Page 269
7.1 The Inner Product Space C^n......Page 274
7.2 Orthogonal Sets of Vectors in C^n......Page 279
7.3 QR Factorization......Page 286
7.3.1 Kung's Algorithm......Page 291
7.3.2.1 The QR Factorization......Page 293
7.4 A Fundamental Theorem of Linear Algebra......Page 295
7.5 Minimum Norm Solutions......Page 299
7.6 Least Squares......Page 302
8.1 Orthogonal Projections......Page 308
8.2 The Geometry of Subs paces and the Algebra of Projections......Page 316
8.3 The Fundamental Projections of a Matrix......Page 326
8.4 Full Rank Factorizations of Projections......Page 330
8.5 Affine Projections......Page 332
8.6 Quotient Spaces (optional)......Page 341
9.1 Eigenstuff......Page 346
9.1.1.1 Eigenvalues and Eigenvectors in MATLAB......Page 354
9.2 The Spectral Theorem......Page 355
9.3 The Square Root and Polar Decomposition Theorems......Page 364
10.1 Diagonalization with Respect to Equivalence......Page 368
10.2 Diagonali.lation with Respect to Similarity......Page 374
10.3 Diagonahzation with Respect to a Unitary......Page 388
10.3.1.1 Schur Triangularization......Page 393
10.4 The Singular Value Decomposition......Page 394
10.4.1.1 The Singular Value Decomposition......Page 402
11.1.1 Jordan Blocks......Page 406
11.1.2 Jordan Segments......Page 409
11.1.2.1 MATLAB Moment......Page 412
11.1.3 Jordan Matrices......Page 413
11.1.3.1 MATLAB Moment......Page 414
11.1.4 Jordan's Theorem......Page 415
11.1.4.1 Generalized Eigenvectors......Page 419
11.2 The Smith Normal Form (optional)......Page 439
12.1 Bilinear Forms......Page 448
12.2 Matrices Associated to Bilinear Forms......Page 454
12.3 Orthogonality......Page 457
12.4 Symmetric Bilinear Forms......Page 459
12.5 Congruence and Symmetric Matrices......Page 464
12.6 Skew-Symmetric Bilinear Forms......Page 467
12.7 Tensor Products of Matrices......Page 469
12.7.1.1 Tensor Product of Matrices......Page 473
A.1 What Is a Scalar?......Page 476
A.2 The System of Complex Numbers......Page 481
A.3.1.4 Commutative Law of Addition......Page 483
A.3.1.9 Existence of Inverses......Page 484
A.4 Complex Conjugation, Modulus, and Distance......Page 485
A.4.2 Basic Facts about Magnitude......Page 486
A.4.3 Basic Properties of Distance......Page 487
A.5 The Polar Form of Complex Numbers......Page 490
A.6 Polynomials over C......Page 497
A.7 Postscript......Page 499
B.1 Introduction......Page 502
B.2 Matrix Addition......Page 504
B.3 Scalar Multiplication......Page 506
B.4 Matrix Multiplication......Page 507
B.5 Transpose......Page 512
B.5.1.1 Matrix Manipulations......Page 519
B.6 Submatrices......Page 520
B.6.1.1 Getting at Pieces of Matrices......Page 523
C.1 Motivation......Page 526
C.2 Defining Determinants......Page 529
C.3.2 The Cauchy-Binet Theorem......Page 534
C.3.3 The Laplace Expansion Theorem......Page 537
C.4 The Trace of a Square Matrix......Page 545
D.1 Spanning......Page 548
D.2 Linear Independence......Page 550
D.3 Basis and Dimension......Page 551
D.4 Change of Basis......Page 555
Index......Page 560
