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Advanced Topics in Linear Algebra
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Advanced Topics in
Linear Algebra
Weaving Matrix Problems through the Weyr Form
KEVIN C. O’MEARA
JOHN CLARK
CHARLES I. VINSONHALER
3
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3
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Library of Congress Cataloging-in-Publication Data
O’Meara, Kevin C.
Advanced topics in linear algebra : weaving matrix problems through the
Weyr Form / Kevin C. O’Meara, John Clark, Charles I. Vinsonhaler.
p. cm. Includes bibliographical references and index.
ISBN 978-0-19-979373-0
1. Algebras, Linear. I. Clark, John. II. Vinsonhaler, Charles Irvin, 1942- III. Title.
QA184.2.O44 2011
512’.5-dc22
2011003565
9
8
7
6
5
4
3
2
1
Printed in the United States of America
on acid-free paper
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DEDICATED TO
Sascha, Daniel, and Nathania
Kevin O’Meara
Austina and Emily Grace Clark
John Clark
Dorothy Snyder Vinsonhaler
Chuck Vinsonhaler
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CONTENTS
Preface xi
Our Style xvii
Acknowledgments xxi
PART ONE: The Weyr Form 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 = C 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
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Contents
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 C[A1 , . . . , Ak ] 297
6.12. ASD for Commuting Triples of Low Order Matrices 301
Biographical Notes on Motzkin and Taussky 307
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Contents
7. Algebraic Varieties 309
7.1. Affine Varieties and Polynomial Maps 311
7.2. The Zariski Topology on Affine n-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
ix
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PREFACE
“Old habits die hard.” This maxim may help explain why the Weyr form has been
almost completely overshadowed by its cousin, the Jordan canonical form. Most
have never even heard of the Weyr form, a matrix canonical form discovered
by the Czech mathematician Eduard Weyr in 1885. In the 2007 edition of
the Handbook of Linear Algebra, a 1,400-page, authoritative reference on linear
algebra matters, there is not a single mention of the Weyr form (or its associated
Weyr characteristic). But this canonical form is as useful as the Jordan form,
which was discovered by the French mathematician Camille Jordan in 1870.
Our book is in part an attempt to remedy this unfortunate situation of a grossly
underutilized mathematical tool, by making the Weyr form more accessible to
those who use linear algebra at its higher level. Of course, that class includes
most mathematicians, and many others as well in the sciences, biosciences, and
engineering. And we hope our book also helps popularize the Weyr form by
demonstrating its topical relevance, to both “pure” and “applied” mathematics.
We believe the applications to be interesting and surprising.
Although the unifying theme of our book is the development and applications
of the Weyr form, this does not adequately describe the full scope of the
book. The three principal applications—to matrix commutativity problems,
approximate simultaneous diagonalization, and algebraic geometry—bring the
reader right up to current research (as of 2010) with a number of open
questions, and also use techniques and results in linear algebra not involving
canonical forms. And even in topics that are familiar, we present some unfamiliar
results, such as improving on the known fact that commuting matrices over an
algebraically closed field can be simultaneously triangularized.
Matrix canonical forms (with respect to similarity) provide exemplars for
each similarity class of square n × n matrices over a fixed field. Their aesthetic
qualities have long been admired. But canonical forms also have some very
concrete applications. The authors were drawn to the Weyr form through a
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Preface
question that arose in phylogenetic invariants in biomathematics in 2003. Prior
to that, we too were completely unaware of the Weyr form. It has been a lot of
fun rediscovering the lovely properties of the Weyr form and, in some instances,
finding new properties. In fact, quite a number of results in our book have
(apparently) not appeared in the literature before. There is a wonderful mix
of ideas involved in the description, derivation, and applications of the Weyr
form: linear algebra, of course, but also commutative and noncommutative
ring theory, module theory, field theory, topology (Euclidean and Zariski),
and algebraic geometry. We have attempted to blend these ideas together
throughout our narrative. As much as possible, given the limits of space, we have
given self-contained accounts of the nontrivial results we use from outside the
area of linear algebra, thereby making our book accessible to a good graduate
student. For instance, we develop from scratch a fair bit of basic algebraic
geometry, which is unusual in a linear algebra book. If nothing else, we claim
to have written quite a novel linear algebra text. We are not aware of any book
with a significant overlap with the topics in ours, or of any book that devotes
an entire chapter to the Weyr form. However, Roger Horn recently informed
us (in September 2009) that the upcoming second edition of the Horn and
Johnson text Matrix Analysis will have a section on the Weyr form in Chapter 3.
Of course, whether our choice of topics is good or bad, and what sort of job we
have done, must ultimately be decided by the reader.
All seven chapters of our book begin with a generous introduction, as do most
sections within a chapter. We feel, therefore, that there is not a lot of point in
describing the chapter contents within this preface, beyond the barest summary
that follows. Besides, the reader is not expected to know what the Weyr form is
at this time.
PART I: THE WEYR FORM AND ITS PROPERTIES
1:
2:
3:
Background Linear Algebra
We do a quick run-through of some of the more important basic
concepts we require from linear algebra, including diagonalization of
matrices, the description of the Jordan form, and desirable features of
canonical matrix forms in general.
The Weyr Form
Here we derive the Weyr form from scratch, establish its basic
properties, and detail an algorithm for computing the Weyr form of
nilpotent matrices (always the core case). We also derive an important
duality between the Jordan and Weyr structures of nilpotent matrices.
Centralizers
The matrices that centralize (commute with) a given nilpotent Jordan
matrix have a known explicit description. Here we do likewise for the
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Weyr form, for which the centralizer description is simpler. It is this
property that gives the Weyr form an edge over its Jordan counterpart
in a number of applications.
The Module Setting
The Jordan form has a known ring-theoretic derivation through
decompositions of finitely generated modules over principal ideal
domains. In this chapter we derive the Weyr form ring theoretically,
but in an entirely different way, by using ideas from decompositions
of projective modules over von Neumann regular rings. The results
suggest that the Weyr form lives in a somewhat bigger universe than its
Jordan counterpart, and is perhaps more natural.
PART II: APPLICATIONS OF THE WEYR FORM
5:
6:
7:
Gerstenhaber’s Theorem
The theorem states that the subalgebra F [A, B] generated by two
commuting n × n matrices A and B over a field F has dimension at
most n. It was first proved using algebraic geometry, but later Barría
and Halmos, and Laffey and Lazarus, gave proofs using only linear
algebra and the Jordan form. Here we simplify the Barría–Halmos
proof even further through the use of the Weyr form in tandem with the
Jordan form, utilizing an earlier duality.
Approximate Simultaneous Diagonalization
Complex n × n matrices A1 , A2 , . . . , Ak are called approximately
simultaneously diagonalizable (ASD) if they can be perturbed
to simultaneously diagonalizable matrices B1 , B2 , . . . , Bk . In this
chapter we attempt to show how the Weyr form is a promising tool
(more so than the Jordan form) for establishing ASD of various
classes of commuting matrices using explicit perturbations. The ASD
property has been used in the study of phylogenetic invariants in
biomathematics, and multivariate interpolation.
Algebraic Varieties
Here we give a largely self-contained account of the algebraic geometry
connection to the linear algebra problems studied in Chapters 5
and 6. In particular, we cover most of the known results on the
irreducibility of the variety C (k , n) of all k-tuples of commuting
complex n × n matrices. The Weyr form is used to simplify some
earlier arguments. Irreducibility of C (k, n) is surprisingly equivalent
to all k commuting complex n × n matrices having the ASD property
described in Chapter 6. But a number of ASD results are known only
through algebraic geometry. Some of this work is quite recent (2010).
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Preface
Our choice of the title Advanced Topics in Linear Algebra indicates that we are
assuming our reader has a solid background in undergraduate linear algebra (see
the introduction to Chapter 1 for details on this). However, it is probably fair
to say that our treatment is at the higher end of “advanced” but without being
comprehensive, compared say with Roman’s excellent text Advanced Linear
Algebra,1 in the number of topics covered. For instance, while some books on
advanced linear algebra might take the development of the Jordan form as one
of their goals, we assume our readers have already encountered the Jordan form
(although we remind readers of its properties in Chapter 1). On the other
hand, we do not assume our reader is a specialist in linear algebra. The book is
designed to be read in its entirety if one wishes (there is a continuous thread),
but equally, after a reader has assimilated Chapters 2 and 3, each of the four
chapters that follow Chapter 3 can be read in isolation, depending on one’s
“pure” or “applied” leanings.
At the end of each chapter, we give brief biographical sketches of one or two
of the principal architects of our subject. It is easy to forget that mathematics has
been, and continues to be, developed by real people, each generation building
on the work of the previous—not tearing it down to start again, as happens
in many other disciplines. These sketches have been compiled from various
sources, but in particular from the MacTutor History of Mathematics web
site of the University of St. Andrews, Scotland [ and I. Kleiner’s A History of Abstract Algebra. Note,
however, that we have given biographies only for mathematicians who are no
longer alive.
When we set out to write this book, we were not thinking of it as a text
for a course, but rather as a reference source for teachers and researchers. But
the more we got into the project, the more apparent it became that parts of
the book would be suitable for graduate mathematics courses (or fourth-year
honors undergraduate courses, in the case of the better antipodean universities).
True, we have not included exercises (apart from a handful of test questions),
but the nature of the material is such that an instructor would find it rather easy
(and even fun) to make up a wide range of exercises to suit a tailored course. As
to the types of course, a number spring to mind:
(1) A second-semester course following on from a first-year graduate
course in linear algebra, covering parts of Chapters 1, 2, 3, and 6.
1. Apart from our background in Chapter 1, there is no overlap in the topics covered in our book
and that of Roman.
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xv
(2) A second-semester course following on from an abstract algebra
course that covered commutative and noncommutative rings,
covering parts of Chapters 1, 2, 3, 4, 5, and 7.
(3) The use of Chapter 4 as a supplement in a course on module theory.
(4) The use of Chapter 7 as a supplement in a course on algebraic
geometry or biomathematics (e.g., phylogenetics).
The authors welcome comments and queries from readers. Please use the
following e-mail addresses:
(Kevin O’Meara)
(John Clark)
(Chuck Vinsonhaler)
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OUR STYLE
Mathematicians are expected to be very formal in their writings. An unintended
consequence of this is that mathematics has more than its share of rather boring,
pedantic, and encyclopedic books—good reading for insomniacs. We have
made a conscious decision to write in a somewhat lighter and more informal
style. We comment here on some aspects of this style, so that readers will know
what to expect. The mathematical content of our arguments, on the other hand,
is always serious.
Some mathematics writers believe that because they have formally spelled
out all the precise definitions of every concept, often lumped together at the
very beginning of a chapter or section, the reader must be able to understand
and appreciate their arguments. This is not our experience. (One first expects
to see the menu at a restaurant, not a display of all the raw ingredients.) And
surely, if a result is stated in its most general form, won’t the reader get an even
bigger insight into the wonders of the concepts? Mistaken again, in our view,
because this may obscure the essence of the result. To complete the trifecta of
poor writing, mathematicians sometimes try to tell the reader everything they
know about a particular topic; in so doing, they often cloud perspective. We
have kept the formal (displayed) statements of definitions to a minimum—
reserved for the most important concepts. We have also attempted to delay the
formal definition until after suitable motivation of the concept. The concept is
usually then illustrated by numerous examples. And in the development proper,
we don’t tell everything we know. In fact, we often invite (even challenge) the
reader to continue the exploration, sometimes in a footnote.
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Our Style
We make no apology for the use of whimsy.1 In our view, there is a place
for whimsy even within the erudite writings of mathematicians. It can help
put a human face on the authors (we are not high priests) and can energize
a reader to continue during the steeper climbs. Our whimsical comments are
mostly reserved for an occasional footnote. But footnotes, being footnotes, can
be skipped without loss of continuity to the story.
We have tried to avoid the formality of article writing in referencing works.
Thus, rather than say “ see Proposition 4.8 (2) and the Corollary on p. 222
of [BAII] ” we would tend to say simply “ see Chapter 4 of Jacobson’s Basic
Algebra II.” Likewise, an individual paper by Joe Blog that is listed in our
bibliography will usually be referred to as “the 2003 paper by Blog,” if there is
only one such paper. The interested reader can then consult the source for more
detail.
What constitutes “correct grammar” has been a source of much discussion
and ribbing among the three authors, prompted by their different education
backgrounds (New Zealand, Scotland, and U.S.A.). By and large, the British
Commonwealth has won out in the debate. But we are conscious, for example,
of the difference in the British and American use of “that versus which,”2 and in
punctuation. So please bear with us.
Our notation and terminology are fairly standard. In particular, we don’t
put brackets around the argument in the dimension dim V of a vector space
V or the rank of a matrix A, rank A. However, we do use brackets if both
the mathematical operator and argument are in the same case. Thus, we write
ker(b) and im(p) for the kernel and image of module homomorphisms b and
p, rather than the visually off-putting ker b and im p. Undoubtedly, there will
be some inconsistencies in our implementation of this policy. An index entry
such as Joe Blog’s theorem, 247, 256, 281 indicates that the principal statement
or discussion of the theorem can be found on page 256, the one in boldface.
This is done sparingly, reserved for the most important concepts, definitions, or
results. Very occasionally, an entry may have more than one boldfaced page to
indicate the most important, but separate, treatments of a topic.
Finally, a word to a reader who perceives some “cheerleading” on our part
when discussing the Weyr form. We have attempted to be even-handed in our
1. In a 2009 interview (by Geraldine Doogue, Australian ABC radio), Michael Palin (of Monty
Python and travel documentary fame, and widely acclaimed as a master of whimsy) was asked
why the British use whimsy much more so than Americans. His reply, in part, was that Britain
has had a more settled recent history. America has been more troubled by wars and civil rights.
Against this backdrop, Americans have tended to take things more seriously than the British.
2. Our rule is “that” introduces a defining clause, whereas “which” introduces a nondefining
clause.
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Our Style
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treatment of the Weyr and Jordan forms (the reader should find ample evidence
of this). But when we are very enthused about a particular result or concept,
we tell our readers so. Wouldn’t life be dull without such displays of feeling?
Unfortunately, mathematics writers often put a premium on presenting material
in a deadpan, minimalist fashion.
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ACKNOWLEDGMENTS
These fall into two groups : (1) A general acknowledgment of those people who
contributed to the mathematics of our book or its publishing, and (2) A personal
acknowledgment from each of the three authors of those who have given moral
and financial support during the writing of the book, as well as a recognition of
those who helped support and shape them as professional mathematicians over
some collective 110 years!
We are most grateful to Mike Steel (University of Canterbury, New Zealand)
for getting us interested in the linear algebra side of phylogenetics, and to
Elizabeth Allman (University of Alaska, Fairbanks) for contributing the section
on phylogenetics in Chapter 6. Many thanks to Paul Smith (University of
Washington, Seattle) for supplying the proof of a Denseness Theorem in
Chapter 7. We also single out Roger Horn (University of Utah, Salt Lake
City) for special thanks. His many forthright, informative comments on an
earlier draft of our book, and his subsequent e-mails, have greatly improved the
final product.
It is a pleasure to acknowledge the many helpful comments, reference
sources, technical advice, and the like from other folk, particularly Pere Ara,
John Burke, Austina Clark, Herb Clemens, Keith Conrad, Ken Goodearl,
Robert Guralnick, John Hannah, John Holbrook, Robert Kruse, James
Milne, Ross Moore, Miki Neumann, Keith Nicholson, Vadim Olshevsky,
Matja Omladiˇc, Bob Plemmons, John Shanks, Boris Shekhtman, Klemen
Šivic, Molly Thomson, Daniel Velleman (Editor of American Mathematical
Monthly), Graham Wood, and Milen Yakimov.
To the four reviewers who reported to Oxford University Press on an earlier
draft of our book, and who made considered, insightful comments, we say thank
you very much. In particular, we thank the two of you who suggested that our
original title The Weyr Form: A Useful Alternative to the Jordan Canonical Form
did not convey the full scope of our book.
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Acknowledgments
Finally, our sincere thanks to editor Phyllis Cohen and her assistant Hallie
Stebbins, production editor Jennifer Kowing, project manager Viswanath
Prasanna from Glyph International, and the rest of the Oxford University Press
(New York) production team (especially the copyeditor and typesetter) for
their splendid work and helpful suggestions. They freed us up to concentrate
on the writing, unburdened by technical issues. All queries from us three
greenhorns were happily and promptly answered. It has been a pleasure working
with you.
From Kevin O’Meara. The biggest thanks goes to my family, of whom I
am so proud: wife Leelalai, daughters Sascha and Nathania, and son Daniel.
They happily adopted a new member into the family, “the book.” Thanks also
to those who fed and sheltered me during frequent trips across the Tasman
(from Brisbane to Christchurch and Dunedin), and across the Pacific (from
Christchurch to Storrs, Connecticut), in connection with the book (or its
foundations): John and Anna-Maree Burke, Brian and Lynette O’Meara, Lloyd
and Patricia Ashby, Chuck and Patty Vinsonhaler, Mike and Susan Stuart, John
and Austina Clark, Gabrielle and Murray Gormack. I have had great support
from the University of Connecticut (U.S.A.) during my many visits over the
last 30 years, particularly from Chuck Vinsonhaler and Miki Neumann. The
University of Otago, New Zealand (host John Clark) has generously supported
me during the writing of this book. Many fine mathematicians have influenced
me over the years: Pere Ara, Richard Brauer, Ken Goodearl, Israel Herstein,
Nathan Jacobson, Robert Kruse (my Ph.D. adviser), and James Milne, to name
a few. I have also received generous support from many mathematics secretaries
and technical staff, particularly in the days before I got round to learning LATEX :
Ann Tindal, Tammy Prentice, Molly Thomson, and John Spain are just four
representative examples. Finally, I thank Gus Oliver for his unstinting service in
restoring the health of my computer after its bouts of swine ’flu.
From John Clark. First and foremost, I’m most grateful to Kevin and Chuck
for inviting me on board the good ship Weyr form. Thanks also to the O’Meara
family for their hospitality in Brisbane and to the Department of Mathematics
and Statistics of the University of Otago for their financial support. Last, but
certainly not least, my love and gratitude to my wife Austina for seeing me
through another book.
From Chuck Vinsonhaler. I am grateful to my wife Patricia for her support,
and thankful for the mathematical and expository talents of my coauthors.
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PART ONE
The Weyr Form and Its Properties
I
n the four chapters that compose the first half of our book, we develop the
Weyr form and its properties, starting from scratch. Chapters 2 and 3 form
the core of this work. Chapter 1 can be skipped by readers with a solid
background in linear algebra, while Chapter 4, which gives a ring-theoretic
derivation of the Weyr form, is optional (but recommended) reading.
Applications involving the Weyr form come in Part II.
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