Preface ........................................................ xi
Our Style .................................................... xvii
Acknowledgments ............................................... xxi
PART ONE: The Weyr Fonn and Its Properties ...................... 1
1 Background Linear Algebra .................................... 3
1.1 The Most Basic Notions .................................. 4
1.2 Blocked Matrices ....................................... 11
1.3 Change of Basis and Similarity ......................... 17
1.4 Diagonalization ........................................ 22
1.5 The Generalized Eigenspace Decomposition ............... 27
1.6 Sylvester's Theorem on the Matrix Equation AX — XB =
С ...................................................... 33
1.7 Canonical Forms for Matrices ........................... 35
Biographical Notes on Jordan and Sylvester ............. 42
2 The Weyr Form ............................................... 44
2.1 What Is the Weyr Form? ................................. 46
2.2 Every Square Matrix Is Similar to a Unique Weyr
Matrix ................................................. 56
2.3 Simultaneous Triangularization ......................... 65
2.4 The Duality between the Jordan and Weyr Forms .......... 74
2.5 Computing the Weyr Form 82 Biographical Note on Weyr ... 94
3 Centralizers ................................................ 96
3.1 The Centralizer of a Jordan Matrix ..................... 97
3.2 The Centralizer of a Weyr Matrix ...................... 100
3.3 A Matrix Structure Insight into a Number-Theoretic
Identity .............................................. 105
3.4 Leading Edge Subspaces of a Subalgebra ................ 108
3.5 Computing the Dimension of a Commutative Subalgebra ... 114
Biographical Note on Frobenius ............................. 123
4 The Module Setting ......................................... 124
4.1 A Modicum of Modules .................................. 126
4.2 Direct Sum Decompositions ............................. 135
4.3 Free and Projective Modules ........................... 144
4.4 Von Neumann Regularity ................................ 152
4.5 Computing Quasi-Inverses .............................. 159
4.6 The Jordan Form Derived Module-Theoretically .......... 169
4.7 The Weyr Form of a Nilpotent Endomorphism:
Philosophy ............................................ 174
4.8 The Weyr Form of a Nilpotent Endomorphism:
Existence ............................................. 178
4.9 A Smaller Universe for the Jordan Form? ............... 185
4.10 Nilpotent Elements with Regular Powers ................ 188
4.11 A Regular Nilpotent Element with a Bad Power .......... 195
Biographical Note on Von Neumann ........................... 197
PART TWO: Applications of the Weyr Form ....................... 199
5 Gerstenhaber's Theorem ..................................... 201
5.1 k-Generated Subalgebras and Nilpotent Reduction ....... 203
5.2 The Generalized Cayley-Hamilton Equation .............. 210
5.3 Proof of Gerstenhaber's Theorem ....................... 216
5.4 Maximal Commutative Subalgebras ....................... 221
5.5 Pullbacks and 3-Generated Commutative Subalgebras ..... 226
Biographical Notes on Cayley and Hamilton .................. 236
6 Approximate Simultaneous Diagonalization ................... 238
6.1 The Phylogenetic Connection ........................... 241
6.2 Basic Results on ASD Matrices ......................... 249
6.3 The Subalgebra Generated by ASD Matrices .............. 255
6.4 Reduction to the Nilpotent Case ....................... 258
6.5 Splittings Induced by Epsilon Perturbations ........... 260
6.6 The Centralizer of ASD Matrices ....................... 265
6.7 A Nice 2-Correctable Perturbation ..................... 268
6.8 The Motzkin-Taussky Theorem ........................... 271
6.9 Commuting Triples Involving a 2-Regular Matrix ........ 276
6.10 The 2-Regular Nonhomogeneous Case ..................... 287
6.11 Bounds on dim С[A1, ..., Ak] .......................... 297
6.12 ASD for Commuting Triples of Low Order Matrices ....... 301
Biographical Notes on Motzkin and Taussky .................. 307
7 Algebraic Varieties ........................................ 309
7.1 Affine Varieties and Polynomial Maps .................. 311
7.2 The Zariski Topology on Affine и-Space ................ 320
7.3 The Three Theorems Underpinning Basic Algebraic
Geometry .............................................. 326
7.4 Irreducible Varieties ................................. 328
7.5 Equivalence of ASD for Matrices and Irreducibility
of C(k, n) ............................................ 339
7.6 Gerstenhaber Revisited ................................ 342
7.7 Co-Ordinate Rings of Varieties ........................ 347
7.8 Dimension of a Variety ................................ 353
7.9 Guralnick's Theorem for C(3, n) ....................... 364
7.10 Commuting Triples of Nilpotent Matrices ............... 370
7.11 Proof of the Denseness Theorem ........................ 378
Biographical Notes on Hilbert and Noether .................. 381
Bibliography .................................................. 384
Index ......................................................... 390
|
