Thirty-three Miniatures: Mathematical and Algorithmic Applications of Linear Algebra

194 Pages·2010·1.99 MB·English

by  Jiri Matousek

STUDENT MATHEMATICAL LIBRARY
Volume 53
Thirty-three
Miniatures
Mathematical and
Algorithmic Applications
of Linear Algebra
Jiˇrí Matoušek
American Mathematical Society

stml-53-matousek-cov.indd 1 4/27/10 11:06 AM
STUDENT MATHEMATICAL LIBRARY
Volume 53
Thirty-three
Miniatures
Mathematical and Algorithmic
Applications of Linear Algebra
Jirˇí Matoušek
American Mathematical Society
Providence, Rhode Island
Editorial Board
Gerald B. Folland Robin Forman Brad G. Osgood (Chair)
2010 Mathematics Subject Classification. Primary 05C50, 68Wxx, 15–01.
For additional information and updates on this book, visit
www.ams.org/bookpages/stml-53
Library of Congress Cataloging-in-Publication Data
Matouˇsek,Jiˇr´ı,1963–
Thirty-threeminiatures: mathematicalandalgorithmicapplicationsoflinear
algebra/Jiˇr´ıMatouˇsek.
p.cm. —(Studentmathematicallibrary;v.53)
Includesbibliographicalreferencesandindex.
ISBN978-0-8218-4977-4(alk.paper)
1.algebras,Linear. I.Title.
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2009053079
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Contents
Preface v
Notation ix
Miniature 1. Fibonacci Numbers, Quickly 1
Miniature 2. Fibonacci Numbers, the Formula 3
Miniature 3. The Clubs of Oddtown 5
Miniature 4. Same-Size Intersections 7
Miniature 5. Error-CorrectingCodes 11
Miniature 6. Odd Distances 17
Miniature 7. Are These Distances Euclidean? 19
Miniature 8. PackingComplete Bipartite Graphs 23
Miniature 9. EquiangularLines 27
Miniature 10. Where is the Triangle? 31
Miniature 11. Checking Matrix Multiplication 35
Miniature 12. Tiling a Rectangle by Squares 39
iii
iv Contents
Miniature 13. Three PetersensAre Not Enough 41
Miniature 14. Petersen,Hoffman–Singleton, and Maybe 57 45
Miniature 15. Only Two Distances 51
Miniature 16. Coveringa Cube Minus One Vertex 55
Miniature 17. Medium-Size Intersection Is Hard To Avoid 57
Miniature 18. On the Difficulty of Reducing the Diameter 61
Miniature 19. The End of the Small Coins 67
Miniature 20. Walking in the Yard 71
Miniature 21. Counting Spanning Trees 77
Miniature 22. In How Many Ways Can a Man Tile a Board? 85
Miniature 23. More Bricks—MoreWalls? 97
Miniature 24. Perfect Matchings and Determinants 107
Miniature 25. Turning a Ladder Overa Finite Field 113
Miniature 26. Counting Compositions 119
Miniature 27. Is It Associative? 125
Miniature 28. The Secret Agent and the Umbrella 131
Miniature 29. Shannon Capacity of the Union: A Tale of Two
Fields 139
Miniature 30. EquilateralSets 147
Miniature 31. Cutting Cheaply Using Eigenvectors 153
Miniature 32. Rotating the Cube 163
Miniature 33. Set Pairsand ExteriorProducts 171
Index 179
Preface
SomeyearsagoIstartedgatheringniceapplicationsoflinearalgebra,
and here is the resulting collection. The applications belong mostly
to the main fields of my mathematical interests—combinatorics, ge-
ometry, and computer science. Most of them are mathematical, in
provingtheorems,andsomeincludecleverwaysofcomputingthings,
i.e., algorithms. The appearance of linear-algebraicmethods is often
unexpected.
AtsomepointIstartedtocalltheitemsinthecollection“minia-
tures”. ThenIdecidedthatinordertoqualifyforaminiature,acom-
plete exposition of a result, with backgroundand everything, should
not exceed four typeset pages (A4 format). This rule is absolutely
arbitrary, as rules often are, but it has some rational core—namely,
this extent can usually be covered conveniently in a 90-minute lec-
ture, the standard length at the universities where I happened to
teach. Then,ofcourse,therearesomeexceptionstotherule,suchas
six-pageminiatures that I just couldn’t bring myself to omit.
The collection could obviously be extended indefinitely, but I
thought thirty-three was a nice enough number and a good point
to stop.
The exposition is intended mainly for lecturers (I’ve taught al-
most all of the pieces on various occasions) and also for students
interested in nice mathematical ideas even when they require some
v
vi Preface
thinking. The material is hopefully class-ready, where all details left
to the readershould indeed be devil-free.
Iassumeabackgroundinbasiclinearalgebra,abitoffamiliarity
with polynomials, and some graph-theoretical and geometric termi-
nology. The sections have varying levels of difficulty, and generally
I have ordered them from what I personally regard as the most ac-
cessible to the more demanding.
I wanted each section to be essentially self-contained. With a
goodundergraduatebackgroundyoucanaswellstartreadingatSec-
tion24. Thisiskindofoppositetoatypicalmathematicaltextbook,
wherematerialisdevelopedgradually,andifonewantstomakesense
ofsomethingonpage123,oneusuallyhastounderstandtheprevious
122pages,or with luck, some suitable 38 pages.
Of course, the anti-textbook structure leads to some boring rep-
etitions and, perhaps more seriously, it puts a limit on the degree of
achievable sophistication. On the other hand, I believe there are ad-
vantagesaswell: Igaveupreadingseveraltextbookswellbeforepage
123, after I realized that between the usually short reading sessions
I couldn’t remember the key definitions (people with small children
will know what I’m talking about).
After several sections the reader may spot certain common pat-
terns in the presented proofs, which could be discussed at great
length,butIhavedecidedtoleaveoutanygeneralaccountsonlinear-
algebraicmethods.
Nothing in this text is original, and some of the examples are
ratherwellknownandappearinmanypublications(including,inafew
cases,otherbooksofmine). Severalgeneralreferencebooksarelisted
below. I’vealsoaddedreferencestotheoriginalsourceswhereIcould
find them. However, I’ve kept the historical notes at a minimum,
andI’veput only alimited effortinto tracingthe originsof the ideas
(apologiestoauthorswhoseworkisquotedbadlyornotatall—please
let me know about such cases).
Iwouldalsoappreciatelearningaboutmistakesandhearingsug-
gestionsof how to improvethe exposition.
Preface vii
Further reading. An excellent textbook is
L. Babai and P. Frankl, Linear Algebra Methods
inCombinatorics(Preliminaryversion2),Depart-
mentofComputerScience,TheUniversityofChi-
cago,1992.
Unfortunately, it has never been published officially. It can be ob-
tained,withsomeeffort,aslecturenotesoftheUniversityofChicago.
Itcontainsseveralofthetopicsdiscussedhere,alotofothermaterial
in a similar spirit, and a very nice exposition of some parts of linear
algebra.
Algebraic graphtheory is treated, e.g., in the books
N. Biggs, Algebraic Graph Theory, 2nd edition,
CambridgeUniversity Press,Cambridge, 1993
and
C. Godsil and G. Royle, Algebraic Graph Theory,
Springer, New York, NY, 2001.
Probabilistic algorithms in the spirit of Sections 11 and 24 are well
explained in the book
R. Motwani and P. Raghavan, Randomized Algo-
rithms, Cambridge University Press, Cambridge,
1995.
Acknowledgments. Forvaluablecommentsonpreliminaryversions
of this booklet, I would like to thank Otfried Cheong, Esther Ezra,
Nati Linial, Jana Maxova´, Helena Nyklova´, Yoshio Okamoto, Pavel
Pata´k,OlegPikhurko,andZuzanaSafernova´,aswellasallotherpeo-
plewhomImayhaveforgottentoincludeinthislist. Thanksalsoto
David Wilson for permission to use his picture of a random lozenge
tilinginMiniature22,andtoJenniferWrightSharpforcarefulcopy-
editing. Finally, I’m grateful to many people at the Department of
Applied Mathematicsofthe CharlesUniversityinPragueandatthe
Institute of TheoreticalComputerScience of the ETH Zurichfor ex-
cellent workingenvironments.

Notation
Mostofthenotationisdefined ineachsectionwhereitisused. Here
are several general items that may not be completely unified in the
literature.
The integers are denoted by Z, the rationals by Q, the reals by
R, and F stands for the q-element finite field.
q
The transpose of a matrix A is written as AT. The elements
of that matrix are denoted by a , and similarly for all other Latin
ij
letters. Vectors are typeset in boldface: v,x,y, and so on. If x is a
vectorinKn,whereKissomefield,x standsfortheithcomponent,
i
so x=(x ,x ,...,x ).
1 2 n
Wewrite(cid:2)x,y(cid:3) forthestandardscalar(orinner)productofvec-
tors x,y ∈ Kn: (cid:2)x,y(cid:3) =x y+x y +···+x y . We also interpret
1 2 2 n n
suchx,y asn×1(single-column)matrices,andthus(cid:2)x,y(cid:3) couldalso
be written as xTy. Further, for x ∈ Rn, (cid:5)x(cid:5) = (cid:2)x,x(cid:3)1/2 is the Eu-
clidean norm (length) of the vectorx.
Graphsaresimple and undirected unless statedotherwise; i.e., a
graph G is regarded as a pair (V,E), where V is the vertex set and
E is the edge set, which is a set of unorderedpairs of elements ofV.
ForagraphG,wesometimeswriteV(G)forthevertexsetandE(G)
for the edge set.
ix