Advanced topics in linear algebra. Weaving matrix problems through the Weyr form

Front matters......Page 1
Half-title......Page 2
Title: Advanced Topics in Linear Algebra: Weaving Matrix Problems through the Weyr Form......Page 4
ISBN: 9780199793730......Page 5
Dedication......Page 6
Contents......Page 8
Preface......Page 12
Acknowledgements......Page 22
PART-ONE: The Weyr Form and Its Properties......Page 24
1. Background Linear Algebra......Page 26
1.1. The Most Basic Notions......Page 27
1.2. Blocked Matrices......Page 34
1.3. Change of Basis and Similarity......Page 40
1.4. Diagonalization......Page 45
1.5. The Generalized Eigenspace Decomposition......Page 50
1.6. Sylvester’s Theorem on the Matrix Equation AX − XB = C......Page 56
1.7. Canonical Forms for Matrices......Page 58
Biographical Notes on Jordan and Sylvester......Page 65
2. The Weyr Form......Page 67
2.1. What Is the Weyr Form......Page 69
2.2. Every Square Matrix Is Similar to a Unique Weyr Matrix......Page 79
2.3. Simultaneous Triangularization......Page 88
2.4. The Duality between the Jordan and Weyr Forms......Page 97
2.5. Computing the Weyr Form......Page 105
Biographical Note on Weyr......Page 117
3. Centralizers......Page 119
3.1. The Centralizer of a Jordan Matrix......Page 120
3.2. The Centralizer of a Weyr Matrix......Page 123
3.3. A Matrix Structure Insight into a Number-Theoretic Identity......Page 128
3.4. Leading Edge Subspaces of a Subalgebra......Page 131
3.5. Computing the Dimension of a Commutative Subalgebra......Page 137
Biographical Note on Frobenius......Page 146
4. The Module Setting......Page 147
4.1. A Modicum of Modules......Page 149
4.2. Direct Sum Decompositions......Page 158
4.3. Free and Projective Modules......Page 167
4.4. Von Neumann Regularity......Page 175
4.5. Computing Quasi-Inverses......Page 182
4.6. The Jordan Form Derived Module-Theoretically......Page 192
4.7. The Weyr Form of a Nilpotent Endomorphism: Philosophy......Page 197
4.8. The Weyr Form of a Nilpotent Endomorphism: Existence......Page 201
4.9. A Smaller Universe for the Jordan Form......Page 208
4.10. Nilpotent Elements with Regular Powers......Page 211
4.11. A Regular Nilpotent Element with a Bad Power......Page 218
Biographical Note on Von Neumann......Page 220
PART-TWO: Applications of the Weyr Form......Page 222
5. Gerstenhaber’s Theorem......Page 224
5.1. k-Generated Subalgebras and Nilpotent Reduction......Page 226
5.2. The Generalized Cayley–Hamilton Equation......Page 233
5.3. Proof of Gerstenhaber’s Theorem......Page 239
5.4. Maximal Commutative Subalgebras......Page 244
5.5. Pullbacks and 3-Generated Commutative Subalgebras......Page 249
Biographical Notes on Cayley and Hamilton......Page 259
6. Approximate Simultaneous Diagonalization......Page 261
6.1. The Phylogenetic Connection......Page 264
6.2. Basic Results on ASD Matrices......Page 272
6.3. The Subalgebra Generated by ASD Matrices......Page 278
6.4. Reduction to the Nilpotent Case......Page 281
6.5. Splittings Induced by Epsilon Perturbations......Page 283
6.6. The Centralizer of ASD Matrices......Page 288
6.7. A Nice 2-Correctable Perturbation......Page 291
6.8. The Motzkin–Taussky Theorem......Page 294
6.9. Commuting Triples Involving a 2-Regular Matrix......Page 299
6.10. The 2-Regular Nonhomogeneous Case......Page 310
6.11. Bounds on dim C[A1, . . . , Ak]......Page 320
6.12. ASD for Commuting Triples of Low Order Matrices......Page 324
Biographical Notes on Motzkin and Taussky......Page 330
7. Algebraic Varieties......Page 332
7.1. Affine Varieties and Polynomial Maps......Page 334
7.2. The Zariski Topology on Affine n-Space......Page 343
7.3. The Three Theorems Underpinning Basic Algebraic Geometry......Page 349
7.4. Irreducible Varieties......Page 351
7.5. Equivalence of ASD for Matrices and Irreducibility of C(k, n)......Page 362
7.6. Gerstenhaber Revisited......Page 365
7.7. Co-Ordinate Rings of Varieties......Page 370
7.8. Dimension of a Variety......Page 376
7.9. Guralnick’s Theorem for C(3, n)......Page 387
7.10. Commuting Triples of Nilpotent Matrices......Page 393
7.11. Proof of the Denseness Theorem......Page 401
Biographical Notes on Hilbert and Noether......Page 404
Bibliography......Page 407
Index......Page 413