Table Of ContentVectoSrp aces
and
Matrices
Rob er tM . Thrall
ProfesosfoM ra thematics
UniversoifMt iyc higan
and
LeonardT ornheim
ResearMcaht hematician
CaliforRneisae arCcohr poration
JohnW iley & SonsI,n c.
New York ·Lon•dSoynd ney
Copyrl&©ht 1 957
by
JohnW iley& SonsI,n c.
AllR ightRse served
Thisbo okor any'J)a rtt hermeuosfnot t
ber eproduceidn a nyf orm without
thewrit ten permission oft hep ublisher.
FIFTH PRINTING,F EBRUARY, 1966
LibraroyfC ontreasC ataloC&a rdN umber: 56-9828
Printeidn t heU niteSdt ateosf A merica
Preface
Thep asts everdaelc adehsa vew itnessae gdr owinign tereisntt h e
theoroyfv ectosrp aceasn dm atriceOsn.e e videnocfet hiisn teriesst
ther eplacemoefnt th et raditicoonuarlso en t heoroyfe quatioansts h e
basirce quirceodu rsien a lgebrianm any collegbeys o neo n matrix
theoryA.s idfer omm anyu seisn v ariopuasrt so fm athematimcas,
tricaensd v ectofirnsd u sei nc ircuainta lysis-(ienl ectreincgailn eering),
quantumme chanicsst,a tistniucmse,r icaanla lyseicso,n ometraincds ,
operatiroensse arch.
Thew ider angoef a pplicatoifov nesc tosrpsa ceasn dm atricbeost h
insidaen do utsiodfep urem athematihcass l edt od evelopmeonft
thes ubjeacttm any differelnetv elsE.a chh ighelre veslh ednse w
lighatn dg ivedse epeirn sigihntt ot heb asifca ctsA.f terh aving
lookeddo wnf romt heh eightthsem athematiciissa onm etimteesm pted
tof eeilm patiewnitt hl owelre vealp proacheHso.w everh,e must
realitzhea tt heb eginnisntgu denwth o isn oty ete quipptoe dc limb
tot heh eighntese dtso a pproatchhe s ubjeactta levewli thihni sg rasp.
Att hel oweslte veelv erythiisnk ge ptc oncretAe m.a triixsr egarded
asa na rrayo fr ea(lo rc omplenxu)m berasn,d t hep ageosf a textbook
writtaetnt hilse vealr eg enerousfillyl ewdi ths ummatiosni gnasp
plietdo s equencoefns u mbersA.t a slighhtilgyh elre vetlh em atrices
andv ectobresc omet heb asiecl emenatlst houognhe s tilmuls to c
casionadlelaywl i tht hec omponentAst.t hilse vevle ctoarrse d efined
asn -tuplaensd,m oreg enertaylp eosf v ectosrp aceasr en otc onsidered.
Ata stihlilg helre vevle ctosrp aceasr ed efineadx iomaticaanldlt yh,e
theoriysd evelopoevde arn a rbitrary (commufitealtdbi,u vtwe i}t hout
recourtsoet het heoroyf g roupwsi tho peratorAst. t heh ighesrt rata
of thisl evemla tricaerse r eplacbeyd lineatrr ansformatainodn s
bilinefaurn ctiotnhse;t heorpyr oceewdist houmte ntioonf n -tuples.
Ats tihlilg helre vetlhse n umericsacla laarrser eplacbeyde lemenotfs
v
VI PREFACE
a rinagn dt het heoroyf v ectosrp acebse comeas c orolltaorg ye neral
theoreambso ugtr ouprsi,n gasn,d a lgebras.
Eachl evecla rriweist hi tm any advantagaensd disadvantages.
Thel owelre vedlosn otg iveen ougihn sigihntt toh eb asifca ctasn d
theh ighelre vealrse t ooa bstrafcotrt heb eginneIrn. t hep resent
textbowoek h avec hosetno p roceseidm ultaneoautst lwyol evelosn,e
concreatned o nea xiomatiTch.u si nt hefi rscth aptewre introduce
n-tuplaensd a bstrafcrto mt hema seto fa xiomfso ra vectosrp ace.
Eachn ewp roperotfya vectosrp acies fi rsdti scusasteo dn el evealn d
thena tt heo therS.o metimtehse c oncreatpep roacphr ecedaensd
sometimietsf ollotwhse a xiomatoince .W e feetlh att hidsu ala p
proachha sm anya dvantageIsti. n trodutcheess t udetnott hee legance
andp owero fm athematirceaals onibnags eodn a seto fa xiomasn d
helpbsr idgteh eg apt hatl iebse tweetnh ep re�emineonfcp er oblem
solvinfgo undi n moste lementaurnyd ergraducaotuer seasn d the
axiomataipcp roacthh atc haractermiuzcehs m odern researicnh
mathematicOsn. t heo thehra nd,t hef requernett urnt oc oncrete
formulatihoenlspk se ept hes tudenfte'esot n s oligdr oundW.e have
trietdo develoupn derstandainndg c onceptugarla spr,a thetrh an
merem anipulation.
Int hee arlcyh aptewres h aves uppliae fda rm ored etaileexdp osi
tiotnh ani sf ounidn m anyt extbookIsth. a sb eeno ure xperietnhcaet
thifsu llneisnds e taiisle xtremeilmyp ortafnotrt hes tudewnhto is
havinhgi sfi rste xperiewnicteh m athematirciaglo rO.n lya ftehre
hasw onh iss purcsa nh ea ffortdo r eplatchees ometimteesd ioduest ails
whichc lincah p roobfy a phrasseu cha s" continuiinnt gh ifsa sh
ion· · ·. " Moreovesru,c hs tatemefnrtesq uengtlloyso sv era ni nduc
tioanr gumenatn,d w e belietvhea atn o bjectoifvt eh icso ursseh ould
be tgoi vteh es tudean tth orouggrho undiinnpg r oobfy i nduction.
Theb ooki sd esignfeodra one-yecaoru rsmee etintgh retei mepse r
week.W e havef oundt hati na one-semesctoeurr soen ec anc over
thefi rstfi vec haptearnsd s omet opicfsr omt hes ixtahn ds eventh
chapters.
Thefi rsttw oc hapteirnst rodutcheec oncepotfsv ectosrp aclei,n ear
transformatainodnm ,a trixT.h et hircdh aptaeprp litehse sceo ncepts
tot hep robleomfs olvisnygs temosfl ineeaqru ationAsl.t houtghhe se
threceh aptearrsec loseklnyi tt,h ei nstrucwthoorw ishetso w ilfiln d
itp ossibtlore e arrantgheeo rdeorf p resentastoia osnt oc omet ot he
solutioofne quatiosnosm ewhaeta rlieIrn. p articuSleacrt io1n.s9 ,
1.102,. 7a,n d2 .8c anb eo mittoerdp ostponed.
PREFACE VII
Chapte4r g iveas s elf-contdaeivneeldo pmeonftt h et heoroyf d e
terminantTsh.e firstf ours ectiocnasn b e skippeidft hes tudents
havea lreadhya da ni ntroducttoid oent erminants.
Chapte5r i si ntendteodi ntrodutchees tudetnott heg enercaoln cept
ofi nvariaanntds erveassa summaryo fw hath asp recedeIdt.a lso
includae bsr ieifn troducttoit ohnet heoroyf s imilarity.
Chapte6r asn d7 p resetnhteu suamla terioanlb ilinaenadrq uadratic
formsa ndo no rthogoenqauli valence.
Thel asfto ucrh aptearrsem orea dvanceidnn aturaen,d t hel eveolf
expositiisos no mewhahti gherT.h efi rspta rto fC hapte8r g ivetsh e
standatrhde oroyf p olynomiianlo sn ei ndeterminaTthee .l assti x
sectiodnesv elotph eg enertahle oroyf s implael gebraeixct ensioofn s
a field.B y removintgh eu suarle stricttoie oxnt ensiwohnisc ha re
agaifine ldwse geta developmewnhti chc anb eu sedn oto nlya sa
foundatfioornG aloitsh eorbyu ta lsfoo rt heg enersailm ilartihteyo ry.
To theb esotf o urk nowledtghei iss t hefi rsetl ementaprrye sentation
oft himsa teriaavla ilaibnlt ee xtbofookr m( inE nglishI)n.C hapter
9 we presetnhte s tandard canfoonrimcfsao lr m atricweist hi ntegral
orw ithp olynomiealle mentTsh.e ser esulatrsea ppliteodd ifferential
equatiownist hc onstacnote fficienatnsd, a lsot o finitelgeyn erated
abeligarno uptso o btaitnh ef undamenttahle oremC.h apte1r0 p re
senttsh es imilartihteyo rwyi tht heu suacla nonicfaolr msI.t i ncludes
applicattioog neso metrayn dt od ifferenteiqaula tionMsa.n y further
applicatairoenc so nsideirnet dh ee xerciisnet sh icsh apter.
Thefi nalc hapteorn lineairn equaliptrieesse nat bsr ieifn troduc
tiotn ot hiism portatnotp iwch,i chh asu ntinlo w beenn eglectiend
elementatreyx tosn m atrix theWoer hy.o pet hawth atw e have given
wilplr ovitdhee s tudewnitt hs ufficiebnatc kgrousnodt hahte c anr ead
bookasn dp aperosn g amet heorayn dl ineparro gramming.
We havec hoseinn t hibso okt ob reako ff ourt reatmeonftg roup
theorjyu ssth orto fa formadle,v elopmeonftt hec oncepotf f actor
group.T hisd ecisiwoans b asedo n thec onvicttihoant i,f w e went
asf ara sf actogrr oupist,w ouldb ed ifficutlota voirda isitnhgel evel
oft hee ntibroeo kt os ucha pointth aitt w ouldh aveb ecomae text
booko n "groupwsi tho peratowrist ha pplicattioom nast ritxh eory."
Ho}Veveri,nc asewsh ermeo reg routph eoriysa vailaobrls eu itabwlee
recommentdh att hei nstrucptooirn otu th ows omeo fo urp roofcsa n
bes implifieTdh.i sh oldess peciailntl hye l asfto ucrh apters.
We wisht oa cknowledtgheeh elpa nds uggestioofnm sa nyf riends
who reada ndt estetdh em anuscriipnta n earlvye rsionI.n cluded
Vlll PREFACE
amongt hesaer eE .A . CameronG,. K. KaliscJh.,E . McLaughlin,
C.J .N esbit.tJA,.. N yswandeHr.,R aiffaG,. L .T hompsoann,d A .W .
Tucker.
ROBERMT. THRALL
LEONARTDO RNHEIM
March 1957
Contents
CHAPTER1 VectoSrp aces
1.1 Introduction 1
1.2S calars 4
1.3C artesisapna ce 8
1.4V ector spaces 10
1.5I ndependeonfcv ee ctors 13
1.6D imensiaonnd b asis 19
17. Isomorphism 21
1.8S ubspaces 24
1.9S umsa ndi ntersecotfis ounbss paces 26
1.1S0u bspacoefs'U n (:-l) 29
CHAPTER2 LineaTrr ansformatiaonnds M atrices
2.1L ineatrr ansformations 32
2.2M atrioxf a linetarra nsformation 36
2.3E xampleosfl inetarra nsformations 41
2.4M ultiplicoaftl iionne tarra nsformations 45
2.5M ultiplicaotfmi aotnr ices 50
2.6S umsa ndp roducbtyss calars 55
2.7S etosf l inetarra nsformaotfia o vnesc tosrp ace 59
2.8S ubmatripcaerst,i timoanterdi ces 63
2.9R ow andc olummna trices 68
CHAPTER3 Systemosf L ineaErq uations
3.1R ank;r owa ndc olumsnp aceosfa matrix 73
3.2S ystemosfl inehaorm ogeneoeuqsu ations 78
3.3 Elementatrrya nsformations 82
3.4R ow-echelmoant rices 86
ix
x CONTENTS
3.5T riangualnadrH ermitmea trices 89
3.6 PropertoifeH se rmitmea trices 93
3.7E lementamrayt rices 94
3.8E quivaleonfcm ea trices 101
3.9N onhomogeneloiunse eaqru ations 106
CHAPTER 4 Determinants
4.1D efinitioofnd eterminant 110
4.2B asipcr opertoifde est erminants 115
4.3P roofosft hep ropertoifde est erminants 118
4.4C lassical deofifdn eitteiromni nLaanptl;a ce expbayn sion
minors 120
4.5D eterminaonfpt rso ducotfss q uarmea trices;
determinantal fcorrri atnekr ion 125
4.6D eterminaonfpt rso ducotfsr ectangmualtarri ces 127
4.7A djo intasn di nversoefss q uarmea tricCersa;m erR'usl e 129
CHAPTER 5 Equivalence Relaantdi Coannso nicaFlo rms
5.1E quivalerneclea tions 132
5.2C anonicfaolr masn di nvariants 134
5.3A liaasn da libi 136
5.4C hangoef b asis 139
5.5S imilaroifmt ayt riceeisg;e nvalues 142
5.6E quivaleanncdes imilaroiftl yi netarra nsformation1s4 7
CHAPTER 6 Functionosf V ectors
6.1B ilinefaorr ms 149
6.2C anonicfaolr mfso rs kew-symmeatnrdis cy mmetric
matrices 152
6.3Q uadratfiocr ms 158
6.4B ilinefaurn ctiodnusa;ls paces 160
6.5B ilinear scalara nfdmu antcrtiicodenussa; l ities 163
6.6Q uadratfiucn ctions 167
67. Hermitifaunn ctions 171
6.8D eterminaanstm su ltilifnuenacrt ions 174
CHAPTER 7 Orthoatonaanld UnitaryE quivalence
7.1. Euclidsepaanca en di nneprr oducts 179
7.2S chwartzi'nse qualdiitsyt,a nacned,a ngle 181
7.3O rthogonality 183
7.4O rthogosnuabls paces 185
7.5O rthogontarla nsformations 185
CONTENTS xi
7.6D iagonalizoafts iyomnm etrmiact rices 187
77. Unitartyr ansformations 191
7.8A djoinofta linetarra nsformation 193
CHAPTER8 StructuorfeP olynomiaRlin gs
8.1R ingasn ds ubrings 196
8.2E xistenacnedu niquenoefst sr anscendeenxtatle nsions1 99
8.3D ivisiaolng oritghrme;a tecsotm mond ivisor 203
8.4F actorizaotfip oonl ynomials 207
8.5A lgebraeixct ensioofan sfi eld 210
8.6C ongruenocfpe o lynomials 213
8.7D irecstu mso fv ectosrp aces 214
8.8I dempoteanntdsd irescutm s 218
8.9D ecompositoifao lng ebras 221
8.1D0e compositoifso inm plael gebreaxitce nsions 224
CHAPTER9 EquivalenocfeM atriceosv era Ring
9.1M atricoevse ra ring 229
9.2E quivaleonfcm ea tricweist pho lynomeilaelm ents 230
9.3M atrices with elienmteengtesr 236
9.4V ectosrp aceosv etrh ei ntegers 238
9.5F initegleyn eratveedc tosrp aceosv ert hei ntegers 240
9.6S ystemosf l inedairff erential ewqiutahct ioonnsst ant
coefficients 244
CHAPTER1 0 SimilariotfyM atrices
10.1M inimumf unction 247
10.2I nvariant subspaces 250
10.3C ayley-HamiTlhteoonr em 256
10.4P rimary linear transformations 258
10.5S imilargietnye,r caals e 263
10.6S egrceh aracteristics 269
107. Pairosf b ilinfeoarrm ss;y stemosfl inedairff erential
equations 271
10.8P airosf q uadratfiocr ms 273
10.9A pplicattioop nrso jectgievoem etry 275
10.1R0o otosf m atrices 279
CHAPTER1 1 LineaIrn equalities
11.1D efinitiaonndsn otation 284
11.2I nequaliatnidec so nvesxe ts 285
