Table Of ContentMatrix Canonical Forms
notational skills and proof techniques
S. Gill Williamson
©S.GillWilliamson2012. Allrightsreserved.
Preface
Thismaterialisarewritingofnoteshandedoutbymetobeginninggraduate
students in seminars in combinatorial mathematics (Department of Mathe-
matics, University of California San Diego). Topics covered in this seminar
were in algebraic and algorithmic combinatorics. Solid skills in linear and
multilinearalgebrawererequiredofstudentsintheseseminars-especially
in algebraic combinatorics. I developed these notes to review the students’
undergraduatelinearalgebraandimprovetheirproofskills. Wefocusedon
a careful development of the general matrix canonical forms as a training
ground.
IwouldliketothankDr.TonyTrojanowskiforacarefulreadingofthismaterial
andnumerouscorrectionsandhelpfulsuggestions. Iwouldalsoliketothank
ProfessorMikeSharpe,UCSDDepartmentofMathematics,forconsiderable
LaTeXtypesettingassistanceandforhisLinuxLibertinefontoptionstothe
newtxmathpackage.
S.GillWilliamson,2012
http://cseweb.ucsd.edu/~gill
3
x
4
CONTENTS
Chapter1. FunctionsandPermutations 7
Algebraicterminology 7
Sets,lists,multisetsandfunctions 15
Permutations 20
Exercises 26
Chapter2. MatricesandVectorSpaces 27
Review 27
Exercises: subspaces 30
Exercises: spanningsetsanddimension 31
Matrices–basicstuff 33
Chapter3. Determinants 41
Elementarypropertiesofdeterminants 43
Laplaceexpansiontheorem 51
Cauchy-Binettheorem 57
Exercises: CauchyBinetandLaplace 63
Chapter4. Hermite/echelonforms 65
Rowequivalence 65
Hermiteform,canonicalformsanduniqueness 71
StabilizersofGL(n,(cid:75));columnHermiteforms 78
FinitedimensionalvectorspacesandHermiteforms 81
Diagonalcanonicalforms–Smithform 86
Chapter5. Similarityandequivalence 91
Determinantaldivisorsandrelatedinvariants 91
Equivalencevs. similarity 96
Characteristicmatricesandpolynomials 99
RationalandJordancanonicalforms. 100
Index 107
5
CHAPTER 1
Functions and Permutations
Algebraicterminology
In this first section, we summarize for reference certain basic concepts in
algebra. Theseconceptsareusefulforthematerialwedevelophereandare
essentialforreadingrelatedonlinesources(e.g.,Wikipedia).
Remark1.1(Basicsetsandnotation). Weusethenotation(cid:78) = {1,2,...}
forthepositiveintegers. Let(cid:78)0 = {0,1,2,...}denotethenonnegativeintegers,
and let (cid:90) = {0,±1,±2,...} denote the set of all integers. Let ×nS (n-fold
CartesianproductofS)bethesetofn-tuplesfromanonemptysetS. Wealso
useSn forthiscartesianproduct. Aslightlymoregeneralnotationistowrite
Sn for this product where n = {1,...,n} and the exponential notation RD
denotesalln-tuples(i.e.,functions1.38)fromD toR (Weusedeltanotation:
δ(Statement) = 1ifStatementistrue,0ifStatementisfalse.)
Semigroup→Monoid→Group: setswithonebinaryoperation
Afunctionw :S2 →S iscalledabinaryoperation. Itissometimesusefulto
writew(x,y)inasimplerformsuchasxwy orsimplyx ·y orevenjustxy.
Totiethebinaryoperationw toS explicitly,wewrite(S,w)or(S,·).
Definition 1.2 (Semigroup). Let (S,·) be a nonempty set S with a binary
operation “·” . If (x ·y) ·z = x · (y ·z), for all x,y, z ∈ S, then the binary
operation“·”iscalledassociativeand(S,·)iscalledasemigroup.Iftwoelements,
s,t ∈ S satisfys ·t =t ·s thenwesays andt commute. Ifforallx,y ∈ S we
havex ·y =y ·x then(S,·)isacommutative(orabelian)semigroup.
Remark1.3(Semigroup). LetS = M2,2((cid:90)e)bethesetof2×2matriceswith
entriesin(cid:90)e = {0,±2,±4,...},thesetofevenintegers. Definew(X,Y) =XY
to be the standard multiplication of matrices (which is associative). Then
(S,w)isasemigroup. Thissemigroupisnotcommutative(alternatively,itisa
noncommutativesemigrouporasemigroupwithnon-commutingelements).
The semigroup of even integers, ((cid:90)e,·), where “·” denotes multiplication of
integers,iscommutative.
7
Associative + Identity = Monoid
Definition1.4(Monoid). Let(S,·)beasemigroup. Ifthereexistsanelement
e ∈S suchthatforallx ∈S,e·x =x ·e =x,thene iscalledanidentityforthe
semigroup. Asemigroupwithanidentityiscalledamonoid. Ifx ∈S andthere
isay ∈S suchthatx ·y =y ·x =e theny iscalledaninverseofx.
Remark1.5(Monoid). Theidentityisunique(i.e.,ifeande(cid:48)arebothidentities
thene =e ·e(cid:48) =e(cid:48)). Likewise,ify andy(cid:48) areinversesofx,theny(cid:48) =y(cid:48)·e =
y(cid:48)·(x ·y) = (y(cid:48)·x)·y =e ·y =y sotheinverseofx isunique. Notethatthis
lastcomputationshowsthatify(cid:48)satisfiesy(cid:48)x =e (y(cid:48)isa“leftinverse”)andy
satisfiesxy =e (y isa“rightinverse”)theny(cid:48) =y. The2×2matrices,M2,2((cid:90)),
withmatrixmultiplicationformamonoid(identityI2,the2×2identitymatrix).
Associative + Identity + Inverses = Group
Definition1.6(Group). Let(S,·)beamonoidwithidentitye andletx ∈S.
Ifthereisay ∈S suchthatx ·y =y·x =e theny iscalledaninverseofx (see
1.4). Amonoidinwhicheveryelementhasaninverseisagroup.
Remark1.7(Group). Commutativegroups,x ·y =y ·x forallx andy,play
animportantroleingrouptheory. Theyarealsocalledabeliangroups. Note
thattheinverseofanelementx inagroupisunique: ify andy(cid:48)areinverses
ofx,theny(cid:48) =y(cid:48)·e =y(cid:48)·(x ·y) = (y(cid:48)·x)·y =e ·y =y (see1.5).
Ring: onesetwithtwointertwinedbinaryoperations
Definition 1.8 (Ring and Field). A ring, R = (S,+,·), is a set with two
binaryoperationssuchthat(S,+)isanabeliangroupwithidentitydenoted
by 0 (“+” is called “addition”) and (S − {0},·) is a semigroup (“·” is called
“multiplication”). Thetwooperationsarerelatedbydistributiveruleswhich
statethatforallx,y,z inS:
(left) z ·(x +y) =z ·x +z ·y and (x +y)·z =x ·z+y ·z (right).
Remark1.9(Notation,specialringsandgroupofunits). Thedefini-
tionofaringassumesonlythat(S,·)isasemigroup(1.2). Thus,aringmaynot
haveamultplicativeidentity. Werefertosuchastructureasaringwithout
anidentity. LetR = (S,+,·)bearing. Theidentityoftheabeliangroup(S,+)
isdenotedby0 (or0)andiscalledthezeroofthering(S,+,·). Ifr ∈S then
R
theinverseofr in(S,+)isdenotedby−r sor +(−r) = (−r)+r = 0. Suppose
8
(S −{0},·)isamonoidwithidentity1 (wesay“R isaringwithidentity1 ”);
R R
itsinvertibleelements(orunits),U(R),formagroup,(U(R),·),with1 asthe
R
group identity. The group (U(R),·), or simplyU(R), is the group of units of
theringR. If(S −{0},·)iscommutativethen(S,+,·)isacommutativering. If
(S −{0},·) is a group (i.e., (U(R),·) = (S −{0},·)) then the ringR is called a
skew-fieldordivisionring. Ifthisgroupisabelianthentheringiscalledafield.
Remark1.10(Basicringidentities). Ifr,s,t areinaring(S,+,·)thenthe
followingbasicidentities(inbraces,plushintsforproof)hold:
(1) {r ·0 = 0·r = 0} Ifx+x =x thenx = 0. Takex =r ·0andx = 0·r.
(2) {(−r)·s =r ·(−s) = −(r ·s)} r·s+(−r)·s = 0 =⇒ (−r)·s = −(r·s).
(3) {(−r)·(−s) =r ·s} Replacer by−r in(2). Notethat−(−r) =r.
Inparticular,if(S−{0},·)hasidentity1 ≡ 1,then(−1)·a = −aforanya ∈S
R
and,takinga = −1,(−1)·(−1) = 1. Itisconvenienttodefiner −s =r +(−s).
Thenwehave(r −s)·t =r ·t −s ·t andt ·(r −s) =t ·r −t ·s:
t ·(r −s) =t ·(r +(−s)) =t ·r +t ·(−s)) = (t ·r +−(t ·s)) =t ·r −t ·s.
Afieldisaring(S,+,·)where(S −{0},·)isanabeliangroup
Remark1.11(Rings,fields,identitiesandunits). The2×2matrices
overtheevenintegers,M2,2((cid:90)e),withtheusualmultiplicationandadditionof
matrices,isanoncommutativeringwithoutanidentity. Thematrices,M2,2((cid:90)),
overallintegers,isanoncommutativeringwithanidentity. Thegroupofunits,
(cid:32) (cid:33)
+1 −1
U(M2,2((cid:90))),isallinvertible2×2integralmatrices.ThematrixP = −2 +3
(cid:32) (cid:33)
3 1
is a unit in M2,2((cid:90)) withP−1 = 2 1 .U(M2,2((cid:90))) is usually denoted by
GL(2,(cid:90)) and is called a general linear group. The ring of 2 × 2 matrices of
(cid:32) (cid:33)
x y
theform wherex andy arecomplexnumbersisaskew-fieldbut
−y x
notafield. Thisskew-fieldisequivalentto(i.e,a“matrixrepresentationof”)
theskewfieldofquaternions(seeWikipediaarticleonquaternions). Themost
importantfieldsforuswillbethefieldsofrealandcomplexnumbers.
Definition1.12(Ideal). Let(R,+,·)bearingandletA ⊆ R beasubsetof
R. If(A,+)isasubgroupof(R,+)thenAisaleftidealifforeveryx ∈ R and
y ∈ A,xy ∈ A. Arightidealissimilarlydefined. IfAisbothaleftandright
idealthenitisatwo-sidedidealor,simply,anideal. If(R,+,·)iscommutative
thenallidealsaretwosided.
9
(cid:32) (cid:33)
x y
Remark1.13(Ideal). ThesetAofallmatricesa = formsasubgroup
0 0
(A,+)of(M2,2((cid:90)),+). ThesubsetAisarightidealbutnotaleftidealofthe
ringM2,2((cid:90)).NotethatR = (A,+,·)isitselfaring. Thisringhaspairsofzero
divisors-pairsofelements(a,b)wherea (cid:44) 0andb (cid:44) 0suchthata·b = 0. For
(cid:32) (cid:33) (cid:32) (cid:33)
0 1 1 1
example,takethepair(a,b)tobea = andb = . Thepair
0 0 0 0
(a,b)is,ofcourse,alsoapairofzerodivisorsinM2,2((cid:90)).
Anotherexampleofanidealisthesetofevenintegers,(cid:90) ,whichisasubset
e
oftheintegers,(cid:90)(which,itisworthnoting,formsaringwithnozerodivisor
pairs). Thesubset(cid:90) ,isanideal(two-sided)in(cid:90). Givenanyintegern (cid:44) 0,the
e
set{k ·n|k ∈ (cid:90)}ofmultiplesofn,isanidealofthering(cid:90)whichwedenote
by(n) =n(cid:90) = (cid:90)n. Suchanideal(i.e.,generatedbyasingleelement,n)in(cid:90)
iscalledaprincipalideal. ItiseasytoseethatallidealsAin(cid:90)areprincipal
ideals,(n),where|n| > 0isminimaloverthesetA. Anothernicepropertyof
integersisthattheyuniquelyfactorintoprimes(uptoorderandsign).
Definition1.14(Characteristicofaring). LetR bearing. Givena ∈ R
and an integern > 0, definena ≡ a +a + ···a where there aren terms in
the sum. If there is an integern > 0 such thatna = 0 for alla ∈ R then the
characteristicofRistheleastsuchn.Ifnosuchnexists,thenRhascharacteristic
zero(seeWikipediaarticle“Characteristic(algebra)”forgeneralizations).
Algebraistshavedefinedseveralimportantabstractionsoftheringofintegers,
(cid:90). Wenextdiscussfoursuchabstractions: integraldomains,principalideal
domains(PID),uniquefactorizationdomains(UFD),andEuclideandomains-
eachmorerestrictivethantheother.
EuclideanDomain =⇒ PID =⇒ UFD
Definition1.15(Integraldomain). Anintegraldomainisacommutativering
withidentity,(R,+,·),inwhichtherearenozerodivisorpairs: pairsofnonzero
elements(a,b)whereab = 0.(See1.13forringwithpairsofzerodivisors.)
Remark1.16(Divisors,units,associatesandprimes). Fornoncommu-
tativerings,anelementa isaleftzerodivisorifthereexistsx (cid:44) 0suchthat
ax = 0(rightzerodivisorssimilarlydefined). LetR beacommutativering. If
a (cid:44) 0andb areelementsofaR,wesaythata isadivisorofb (ora dividesb),
a|b, ifthereexistsc suchthatb = ac. Otherwise,a doesnotdivideb,a (cid:45) b.
Note that ifb = 0 and a (cid:44) 0 then a|b becauseb = a0 (c = 0). Thus a|0
ora divides0. (Thetermzerodivisorsof1.15referstopairs(a,b)ofnonzero
elements and is not the same as “a is a divisor of 0” or “a divides 0”.) An
elementu inR−{0}isaninvertibleelementoraunitofR ifu hasaninversein
(R−{0},·). Theunitsformagroup,U(R)(1.9). Forcommutativerings,ab =u,
10
