Finitely Generated Abelian Groups and Similarity of Matrices over a Field

394 Pages·2012·2.1 MB·English

by  Christopher Norman

Springer Undergraduate Mathematics Series
AdvisoryBoard
M.A.J.ChaplainUniversityofDundee,Dundee,Scotland,UK
K.ErdmannUniversityofOxford,Oxford,England,UK
A.MacIntyreQueenMary,UniversityofLondon,London,England,UK
E.SüliUniversityofOxford,Oxford,England,UK
M.R.TehranchiUniversityofCambridge,Cambridge,England,UK
J.F.TolandUniversityofBath,Bath,England,UK
Forfurthervolumes:
www.springer.com/series/3423

Christopher Norman
Finitely Generated
Abelian Groups and
Similarity of Matrices
over a Field
ChristopherNorman
formerlySeniorLecturerinMathematics
RoyalHolloway,UniversityofLondon
London,UK
ISSN1615-2085 SpringerUndergraduateMathematicsSeries
ISBN978-1-4471-2729-1 e-ISBN978-1-4471-2730-7
DOI10.1007/978-1-4471-2730-7
SpringerLondonDordrechtHeidelbergNewYork
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To Lucy,Timand Susie

Preface
Whoisthisbookfor?Thetargetreaderwillalreadyhaveexperiencedafirstcoursein
linear algebra covering matrix manipulation, determinants, linear mappings, eigen-
vectors and diagonalisation of matrices. Ideally the reader will have met bases of
finite-dimensional vector spaces, the axioms for groups, rings and fields as well as
some set theory including equivalence relations. Some familiarity with elementary
numbertheoryisalsoassumed,suchastheEuclideanalgorithmforthegreatestcom-
mondivisoroftwointegers,theChineseremaindertheoremandthefundamentalthe-
orem of arithmetic. In the proof of Lemma6.35 it is assumed that the reader knows
how to resolve a permutation into cycles. With these provisos the subject matter is
virtuallyself-contained.Indeedmanyof thestandard facts of linear algebra,suchas
themultiplicativepropertyofdeterminantsandthedimensiontheorem(anytwobases
of the same finite-dimensional vector space have the same number of vectors), are
proved in a more general context. Nevertheless from a didactic point of view it is
highlydesirable,ifnotessential,forthereadertobealreadyfamiliarwiththesefacts.
Whatdoesthebookdo?Thebookisintwoanalogouspartsandisdesignedtobe
a second course in linear algebra suitable for second/third year mathematics under-
graduates, or postgraduates. The first part deals with the theory of finitely generated
(f.g.) abelian groups: the emerging homology theory of topologicalspaces was built
onsuchgroupsduringthe1870sandmorerecentlytheclassificationofellipticcurves
hasmadeuseofthem.Thestartingpointoftheabstracttheorycouldn’tbemorecon-
creteifittried!Rowandcolumnoperationsareappliedtoanarbitrarymatrixhaving
integerentrieswiththeaimofobtainingadiagonalmatrixwithnon-negativeentries
suchthatthe(1,1)-entryisadivisorofthe(2,2)-entry,the(2,2)-entryisadivisorof
vii
viii Preface
the(3,3)-entry,andsoon;adiagonalmatrixofthistypeissaidtobeinSmithnormal
form (Snf) after the 19th century mathematician HJS Smith. Using an extension of
theEuclideanalgorithmitisshowninChapter1thattheSnfcanbeobtainedwithout
resorttoprimefactorisation.InfacttheexistenceoftheSnfisthecornerstoneofthe
decompositiontheory.
Free abelian groups of finite rank have Z-bases and behave in many ways like
finite-dimensional vector spaces. Each f.g. abelian group is best described as a quo-
tientgroupofsuchafreeabeliangroupbyasubgroupwhichisnecessarilyalsofree.
InChapter2sometimeisspentontheconceptofquotientgroupswhichnostudent
initiallyfindseasy,butluckilyinthiscontextturnsouttobelittlemorethanworking
moduloagiveninteger.Thequotientgroupsarisinginthiswayarespecifiedbymatri-
cesoverZandthetheoryoftheSnfisexactlywhatisneededtoanalysetheirstructure.
PuttingthepiecestogetherinChapter3eachf.g.abeliangroupisseentocorrespond
toasequenceofnon-negativeintegers(itsinvariantfactors)inwhicheachintegerisa
divisorofthenext.Thesequenceofinvariantfactorsofanf.g.abeliangroupencapsu-
latesitsproperties:twof.g.abeliangroupsareisomorphic(abstractlyidentical)ifand
onlyiftheirsequencesofinvariantfactorsareequal.Sobroadly,apartfromimportant
side-issuessuchasspecifyingtheautomorphismsofagivengroupG,thisistheendof
thestoryasfarasf.g.abeliangroupsareconcerned!Neverthelesstheseside-issuesare
thoroughlydiscussedinthetextandthroughnumerousexercises;completesolutions
toallexercisesareontheassociatedwebsite.
Inthesecondpartofthebookthering Z ofintegersisreplacedbythering F[x]
ofpolynomialsoverafieldF.Suchpolynomialsbehaveinthesamewayasintegers
andinparticulartheEuclideanalgorithmcanbeusedtofindthegcdofeachpairof
them. In Chapter 4 the theory of the Smith normal form is shown to extend, almost
effortlessly, to matrices over F[x], the non-zero entries in the Snf here being monic
(leadingcoefficient1)polynomials.Towhatend?Aquestionwhichoccupiescentre
stageinlinearalgebraconcernst×t matricesAandB overafieldF:isthereasys-
tematic method of finding, where it exists, an invertible t ×t matrix X over F with
XA=BX?Should X existthen A and B =XAX−1 arecalledsimilar.Theanswer
tothequestionposedaboveisaresoundingYES!Thesystematicmethodamountsto
reducingthematricesxI−AandxI−B,whicharet×t matricesoverF[x],totheir
Smithnormalforms;iftheseformsareequalthenAandB aresimilarandX canbe
foundbyreferringbacktotheelementaryoperationsusedinthereductionprocesses;
if these forms are different then A and B are not similar and X doesn’t exist. The
matrix xI −A should be familiar to the reader as det(xI −A) is the characteristic
polynomial of A. The non-constant diagonal entries in the Snf of xI −A are called
the invariant factors of A. It is proved in Chapter 6 that A and B are similar if and
onlyiftheirsequencesofinvariantfactorsareequal.Thetheoryculminatesinthera-
tionalcanonicalform(rcf)ofAwhichisthesimplestmatrixhavingthesameinvariant
factorsasA.It’ssignificantthatthercfisobtainedinaconstructiveway;inparticular
thereisnorelianceonfactorisationintoirreduciblepolynomials.
Preface ix
The analogy between the two parts is established using R-modules where R is a
commutative ring. Abelian groups are renamed Z-modules and structure-preserving
mappings(homomorphisms)ofabeliangroupsareZ-linearmappings.Theterminol-
ogy helps the theory along: for instance the reader comfortable with 1-dimensional
subspaces should have little difficulty with cyclic submodules. Each t ×t matrix A
over a field F gives rise to an associated F[x]-module M(A). The relationship be-
tween A and M(A) is explained in Chapter 5 where companion matrices are intro-
duced. Just as each finite abelian group G is a direct sum of cyclic groups, so each
matrixA,asabove,issimilartoadirectsumofcompanionmatrices;thepolynomial
analogueoftheorder|G|ofGisthecharacteristicpolynomialdet(xI −A)ofA.
The theory of the two parts can be conflated using the overarching concept of a
finitelygeneratedmoduleoveraprincipalidealdomain,whichisthestancetakenby
severaltextbooks.AnexceptionisRings,ModulesandLinearAlgebrabyB.Hartley
andT.O.Hawkes,ChapmanandHall(1970)whichopenedmyeyestothebeautyof
the analogy explained above. I willingly acknowledge the debt I owe to this classic
exposition. The two strands are sufficiently important to merit individual attention;
neverthelessIhaveadoptedproofswhichgeneralisewithoutmaterialchange,thatof
theinvariancetheorem3.7beingacaseinpoint.
Mathematically there is nothing new here: it is a rehash of 19th and early 20th
century matrix theory from Smith to Frobenius, ending with the work of Shoda on
automorphisms. However I have not seen elsewhere the step-by-step method of cal-
culatingthe matrix Q described in Chapter 1 though it is easy enoughonce one has
stumbledonthebasicidea.Thebookisanexpansionofmaterialfromalecturecourse
IgaveintheUniversityofLondon,offandon,overa30yearperiodtoundergradu-
atesfirstatWestfieldCollegeandlatterlyatRoyalHolloway.Livelystudentsforced
me to rethink both theory and presentation and I am grateful, in retrospect, to them.
Dr. W.A. Sutherland read and commented on the text and Dr. E.J. Scourfield helped
withthenumbertheoryinChapter3;Ithankboth.Anyerrorswhichremainaremy
own.
FinallyIhopethebookwillattractmathematicsstudentstowhatisundoubtedlyan
importantandbeautifultheory.
London,UK ChristopherNorman