Advanced Topics in Linear Algebra Weaving Matrix Problems through the Weyr Form

by O'Meara, Kevin; Clark, John; Vinsonhaler, Charles

• ISBN13: 9780199793730
• ISBN10: 0199793735
• Format: Hardcover
• Copyright: 2011-09-16
• Publisher: Oxford University Press

Summary

Advanced Topics in Linear Algebra presents, in an engaging style, novel topics linked through the Weyr matrix canonical form, a largely unknown cousin of the Jordan canonical form discovered by Eduard Weyr in 1885. The book also develops much linear algebra unconnected to canonical forms, that has not previously appeared in book form. It presents common applications of Weyr form, including matrix commutativity problems, approximate simultaneous diagonalization, and algebraic geometry, with the latter two having topical connections to phylogenetic invariants in biomathematics and multivariate interpolation. The Weyr form clearly outperforms the Jordan form in many situations, particularly where two or more commuting matrices are involved, due to the block upper triangular form a Weyr matrix forces on any commuting matrix. In this book, the authors develop the Weyr form from scratch, and include an algorithm for computing it. The Weyr form is also derived ring-theoretically in an entirely different way to the classical derivation of the Jordan form. A fascinating duality exists between the two forms that allows one to flip back and forth and exploit the combined powers of each. The book weaves together ideas from various mathematical disciplines, demonstrating dramatically the variety and unity of mathematics. Though the book's main focus is linear algebra, it also draws upon ideas from commutative and noncommutative ring theory, module theory, field theory, topology, and algebraic geometry. Advanced Topics in Linear Algebra offers self-contained accounts of the non-trivial results used from outside linear algebra, and lots of worked examples, thereby making it accessible to graduate students. Indeed, the scope of the book makes it an appealing graduate text, either as a reference or for an appropriately designed one or two semester course. A number of the authors' previously unpublished results appear as well.

Author Biography

Kevin C. O'Meara has taught and researched broadly within algebra, based mostly at the University of Canterbury, New Zealand, but with many visits to the University of Connecticut, USA. Linear algebra has been a recurring theme in much of his work, often in novel settings. John Clark's research interests are in the theory of rings and modules, starting with his Ph.D. written under the supervision of Christian Jensen of the University of Copenhagen. Ring theory has continued to fascinate and surprise him during these last forty years. Charles I. Vinsonhaler has more than 100 publications, most in algebra, a few in actuarial science and mathematics education. He coauthored a small book on problem solving with Tom DeFranco. He has held a number of visiting positions, including the University of Canterbury, New Zealand, where he was an Erskine Fellow.

Table of Contents

Preface	p. xi
Our Style	p. xvii
Acknowledgments	p. xxi
The Weyr Form and Its Properties	p. 1
Background Linear Algebra	p. 3
The Most Basic Notions	p. 4
Blocked Matrices	p. 11
Change of Basis and Similarity	p. 17
Diagonalization	p. 22
The Generalized Eigenspace Decomposition	p. 27
Sylvester's Theorem on the Matrix Equation AX -XB = C	p. 33
Canonical Forms for Matrices	p. 35
Biographical Notes on Jordan and Sylvester	p. 42
The Weyr Form	p. 44
What Is the Weyr Form?	p. 46
Every Square Matrix Is Similar to a Unique Weyr Matrix	p. 56
Simultaneous Triangularization	p. 65
The Duality between the Jordan and Weyr Forms	p. 74
Computing the Weyr Form	p. 82
Biographical Note on Weyr	p. 94
Centralizers	p. 96
The Centralizer of a Jordan Matrix	p. 97
The Centralizer of a Weyr Matrix	p. 100
A Matrix Structure Insight into a Number-Theoretic Identity	p. 105
Leading Edge Subspaces of a Subalgebra	p. 108
Computing the Dimension of a Commutative Subalgebra	p. 114
Biographical Note on Frobenius	p. 123
The Module Setting	p. 124
A Modicum of Modules	p. 126
Direct Sum Decompositions	p. 135
Free and Projective Modules	p. 144
Von Neumann Regularity	p. 152
Computing Quasi-Inverses	p. 159
The Jordan Form Derived Module-Theoretically	p. 169
The Weyr Form of a Nilpotent Endomorphism: philosophy	p. 174
The Weyr Form of a Nilpotent Endomorphism: Existence	p. 178
A Smaller Universe for the Jordan Form?	p. 185
Nilpotent Elements with Regular Powers	p. 188
A Regular Nilpotent Element with a Bad Power	p. 195
Biographical Note on Von Neumann	p. 197
Applications of the Weyr Form	p. 199
Gerstenhaber's Theorem	p. 201
K-Generated Subalgebras and Nilpotent Reduction	p. 203
The Generalized Cayley-Hamilton Equation	p. 210
Proof of Gerstenhaber's Theorem	p. 216
Maximal Commutative Subalgebras	p. 221
Pullbacks and 3-Generated Commutative Subalgebras	p. 226
Biographical Notes on Cayley and Hamilton	p. 236
Approximate Simultaneous Diagonalization	p. 238
The Phylogenetic Connection	p. 241
Basic Results on ASD Matrices	p. 249
The Subalgebra Generated by ASD Matrices	p. 255
Reduction to the Nilpotent Case	p. 258
Splittings Induced by Epsilon Perturbations	p. 260
The Centralizer of ASD Matrices	p. 265
A Nice 2-Correctable Perturbation	p. 268
The Motzkin-Taussky Theorem	p. 271
Commuting Triples Involving a 2-Regular Matrix	p. 276
The 2-Regular Nonhomogeneous Case	p. 287
Bounds on dim C[Ai.....Ak]	p. 297
ASD for Commuting Triples of Low Order Matrices	p. 301
Biographical Notes on Motzkin and Taussky	p. 307
Algebraic Varieties	p. 309
Affine Varieties and Polynomial Maps	p. 311
The Zariski Topology on Affine n-Space	p. 320
The Three Theorems Underpinning Basic Algebraic Geometry	p. 326
Irreducible Varieties	p. 328
Equivalence of ASD for Matrices and Irreducibility of C(c, n)	p. 339
Gerstenhaber Revisited	p. 342
Co-Ordinate Rings of Varieties	p. 347
Dimension of a Variety	p. 353
Guralnick's Theorem for C (3, n)	p. 364
Commuting Triples of Nilpotent Matrices	p. 370
Proof of the Denseness Theorem	p. 378
Biographical Notes on Hilbert and Noether	p. 381
Bibliography	p. 384
Index	p. 390
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