thoery of matrices mac duffee

7/28/2019 Thoery of Matrices Mac Duffee

1/137Publ

icDomain,Google-digitized

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The theory of matrices / by C.C. MacDuffee.

MacDuffee, Cyrus Colton, 1895-

New York : Chelsea , 1956.

http://hdl.handle.net/2027/mdp.49015001327999

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nt ibrary

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.

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E D R MA TH E MA T K

G E N Z G B I E T

B E N V O N D R C H R I FT E I TUN G

L B L A TT F R MA T H E MA T K "

N D

irst dit ion

UB L I S H I N G C O MPA N Y

E W O R K


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HE U N I T D TA T S O F A M R I C A


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maticalabstractionunderlyingmany

s.Thusbilinearand uadraticforms,linear

rcomple systems),linearhomogeneoustrans-

orfunctionsarevariousmanifestationsof

anchesofmathematicsasnumbertheory,

uations,continuedfractions,pro ective

eofcerta inportionsof thissub ect. ndeed,

propertiesofmatriceswerefirstdiscovered

ularapplication,andnotuntilmuchlater re-

y.

hinthescopeof thisboo togiveacompletely

theory,noris itintendedtoma eitan

esub ect. thasbeenthedesireof thew riter

rectionsinwhichthetheoryleads sothatthe

aysee itse tent. Whilesomeattempthas

parts ofthetheory,ingeneralthe material

sfoundintheliterature,the topicsdiscussed

hiche tensiveresearchhasta enplace.

heoremsa briefandelegantproofhas

tishopedthat mostofthesehavebeen

andthatthe readerwillderiveasmuchplea-

did thewriter.

ueDr. aurens arleB ushforacrit ical

ee.


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dDeterminants1

deredsets1

rices5

cands ew matrices5

nts8

0

nors12

antsofhigherdimension15

utativesystems1

ce uation17

tion17

uation17

inimume uation20

ts22

4

acterist icroots25

ofunitarymatrices28

n t eg r al M a tr i c es 2 9

sina principalidealring29

odularmatrices31

s31

sors35

37

4 0

r ic e s 40

delementarydivisors43

ri 44

5

fmatrices48

ormations50

sina principalidealring51

ntegralelements54

sina field5

aca lly closedf ie ld 0

es 2


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8

tsinaf ield 9

stic73

le uivalence75

and orthogonalmatrices78

n of m at r ic e s 81

product81

power-matrices85

u at io ns 8 9

uation89

9 4

tion95

ces97

s97

9

ntsarefunctionsofcomple variables..101

alsofmatrices102

er 104

ts104

0

finiteorder 10

s108

numerablenumberofrowsandcolums. 110


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dDeterminants.

neara lgebra2tofordernoveraf ie ld% is

numbers , / , y , . . . andthreeoperations,

t ion( )andsca larmultiplicationsuchthat

, a a r e un i u e ly d e fi n ed n u mb e rs o f 2 1, w h er e a i s

hat additioniscommutativeandasso-

tionisdistributivewithrespect toaddi-

it isassumedthat

( ab ) , a ) H b p) = (a b o rf ) ,

+ b


13/137

ndDeterminants.

braisassociative,then

) e = e, (e e ) .( ' , , i, 2 , )

mbersci)iaresub ecttothen conditions

. = Z k C i k c , r. ( , /, f, s = l, 2n )

eredsetofnumbers

alarmultiplicationof suchsetsaredefinedby

, r ) = (Z k c t r C , s ) ,

k R k .

eundermultiplicationinthe samemanner

1. f thesetsR tarenotlinearly independent

t obtainedbyborderingR tabovewitha

e le ftw ithdri( ronec er s8), are linearly inde-

icwiththeei undermultiplication2.

sis definedbytheidentity

c ) ,

sgivearepresentationofthe algebra3.

hesameyearthatPo incare snotew as

ote: ThePeirces(subse uently to1858)

alizationofH amilton stheory,andhad

ffect thatprobablyallsystemsofalgebraical

ssociativelawof multiplicationwouldbeeven-

withlineartransformationsofschemata

epresentation....That suchmustbethe

sert,butit isverydifficulttoconceivehow

econsiderationsof 2suggestthat

2overgcan besodefinedthateveryalgebra

somorphicwithoneof itspropersub-algebras5.

lassevonMatrizen, p. 59. B erlin1901.

undihreZ ahlentheorie , p. 35.

. R . A c a d. c i ., P a ri s V o l . 9 9 ( 18 8 4) p p . 74 0 7 4 2.

hm .G es . W is s. P ra gu e (1 88 7) p p. 1 6 1 8 . t ud y, . :

V o l . 4 (1 9 04 ) 1 0 .

. M a th . V o l . ( 1 88 4 ) pp . 2 70 2 8 .

erneA lgebraV o l. p. 37. B erlin1930.


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dDeterminants.

hetotalmatric algebra9ftofordern2 over

ets ofn2numbereachofthe type

operationsandpostulates:

dB = (brs) aree ualifandonly ifar,= brs

sdefinedby

+ b ) .

nabeliangroup withrespecttoaddition,

hee lementsof thering9 . The identity set,

will bedenotedby0.

tionisdefinedby

mn multiplicationof thesets. Theproduct

O A = A O = 0 f o r e v er y A .

ociative,sincemultiplicationinfitis as-

Z u ( ^ ) ^

A ( B C ) .

vewithrespect toaddition,forthe

( a H + i ) cu )

A C + B C .

B ) = C A + C B .

tricalgebraoforder n2overaring 9 is

sconnectionbyspecializing9 , formulti-

allynon-commutativeevenwhen9 isacom-

erseastomultiplicationofA = | = 0doesnot

ingw ithoutdiv isorsofzero . f , how ever,

ite lement1, thenthematri = (


15/137

dDeterminants.

d co lumns, B nhas7row sand co lumns,

sand co lumns. t isreadilyverifiedthat

r iB i s ,( r, s, = 1 , 2)

c s / 1 < ; . , Z ? i . a r e mu l ti p li e d r o w by c o lu m n .

onsofthematricesintobloc s,providedthe

dinthesamew aysastheco lumnsofA .

ray tomeananorderedsetof e lements

e ua lif andonly ifeachconsistsof thesame

correspondinge lementsaree ua l. f

be arrangedintheformof arectangle.Under

sumorproduct oftwoarraysmayhave

ntheyarematricesor asinthelast paragraph

tfrommatrices,but nosuchoperationsare

array.

ceofanarrayof 2elements, butit ismuch

emberofa totalmatricalgebraforwhich

andmultiplicationaredefined.Theimportance

sfromthe rulesofcombinationofmatrices,

ayberepresentedbys uarearraysisin-

ly remar ed: Oneof thef irsto f suche -

relyuncalledfor,especiallyin ngland,

w ordmatri )ta etheplacea lreadysatisfactorily

.H owsatisfactorythiswaswillbe readily

h te t b oo s o f d et e rm i na n ts l i e c o tt s 1 . A

dcertainlyaddfurthertohis creditifin

hemademanifestbypreceptande amplean

ngineachother scompanythetermsarray,

edona consistentterminology.Theword

sedby y lvester3todenotearectangulararray

canbeformed.Theconceptof amatri as

risduein essencetoH amiltonbutmoredirectly

amongothers, uses matri " f o rarectangular

oramemberofamatrica lgebra5. B utthees-

hateletagrees,istodifferentiatetheconcepts.

atiseonthetheoryofdeterminants. C ambridge1880.

o y . o c . . A f r ic a V o l . 1 8 I ( 1 92 9 ) pp . 2 19 - 22 7 .

. V o l . 37 ( 1 85 0 ) pp . 3 3 3 7 0 - C o l l. W o r s V o l .

ndonPhil. oc. V o l. 148(1858)pp. 17-37-C o ll. Wor s

.

upesabeliensf inis. Paris1924.


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dDeterminants.

atrices. A matri o f thetype

]

edefinitionsof additionandmultiplication

+ i h, h, . .. , ] = f t + h, 2 + l 2, .. ., n + l n ,

1, l2 ,. . ., l ] = ft l1 , 2 l2 ,. . . n ln .

o fw hosediagona le lementsaree ua lisca lled

, , ... , ] , th en k + l = k + i , k S { = k l ,

atricesofconstitutea subringofW iso-

f or * a nd A f or k A . f 9 1 is a r in g wi th

/ .

sacommutativeringwithunit element,andif

e ry X i n W l , t he n A i s s c a la r .

s

, thereresults

? drpdi, la i,, a rp8, = drpats. (r, s, p, = 1, 2, . .. , n)

p t h i s gi v es a r p = 0 , wh i le f o r = s a n d r = p

s s.

cands ew matrices. Thematri A T= ( sr)

,)bychangingrowstocolumnsis calledthe

matri suchthat T= isca lledsymmetric,

T = Q i s c a l l ed s e w 3 (s e w -s y mm e tr i c, o r

M a th . V o l . ( 1 88 4 ) pp . 2 70 - 28 - C o l l . W o r s

4.

differentnotationsforthetransposehavebeenused,

, A 1 , , A . T h e pr e se n t no t at i on i s i n e e pi n g wi t h a sy s te m at i c

mayfind favor.

e w. M at h . V o l . 32 ( 1 84 ) p p . 11 9 1 2 3 - C o l l. W o r s

3 3 . a g ue r re : . c o le p o ly t ec h n. V o l . 2 5 (1 8 7 ) p p. 2 1 5 to

vr es V o l . p p. 2 28 -2 33 .


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dDeterminants.

K ) T= A T + B T + r T.

sacommutativering(A B ... )T= T...B TA T.

i ) T = C Z & i r , i ) = B T A T . T he g e ne r al t h eo r em

ringinwhich2 = a issolvableforeverya,

er91is-uni ue lye pressibleasasumofasymmetric

, T = , Q T = -0 .

T= -Q sothat = A + A * ,Q = - .

it ispossibleto formasymmetricmatri and

heabovemanner.

R beana lgebraof - rowedmatricesw ith

sdesirableto haveassociatedwithevery

mber (A )o f% w hichw ouldservethepurposeof

dwouldbe attainedbyfindingascalar

lementsarsofageneralmatri A suchthat

) isanon-constantrationa lintegra lfunction

esuchthat

) (B ) .

( 2) b y t a i n g B = t h at ( A ) = { A ) 9 ( 1) ,

c on s ta n t, ( 1 ) = 1 . H B = 0 , (2 ) g iv e s ( 0 )

a in b e ca u se ( A ) i s n o t c on s ta n t, ( 0 ) = 0 .

i n st a nc e , le t

( t)

7 , T( 1 /< ) = / .

) ) = 1 , i t fo l lo w s th a t ( W ( t )) i s i n de p en d en t

mberof theabovee uationwouldbeof

e (W(t))hasforeveryva lueof tthesame

0, namely


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Determinants.

( V ) = i , ( T = ( T 2 ). i nc e ( T( t) ) 0( ( 1/ O )

esamedegreeX intthat0(7(1/ / )) isin1// , it

tbeamonomiala^ t7 in/ . ince (T(i))= 1,

vo etheminimumprinciple(1)andassumethat

fedbyactua llyobta iningunderthisrestrict iona

enre uirements.Then

1

l). Hence

r i , 9 ( TA ) = t d (A ) s o t ha t

sa po lynomia linthee lementsw hich

rinthe elementsofeachrow.

{ A ) , i t fo ll ow s th at

gessignw hentw orowsarepermuted.

ststatedwerecalledbyW eierstrassthe

fadeterminant. B y1 .

add, whenceby2 ,

+ eA , A i ... A J l A , fl 2A , < * n h = 0 .

/ > , & , . . .A is0if tw osubscriptscoincide, w hile if

e al l d is t in c t, i e i, i ,. .. a c co r di n g as

&

( 1 ) = i , i t f ol l ow s t ha t b 1, 2 = 1 . H e n c e

A ,A , ... A . 1 A 1 2 A ,

overa llpermutations(hl, hi, . . . , h )o f

, ^ is1or 1accordingasthepermutation

. H ense l: . re ineangew . Math. V o l. 159(1928)

r : Z u r D et e rm i na n te n th e or i e. 1 8 8 1 8 87 W e r e

2 8 . r o ne c e r : V o r le s un g en u b er d i e Th e or i e de r D et e r-

t. se . Teubner1903.


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dDeterminants.

)satisfiesthedemandsofH enselwillfollow

ptheentiretheoryofdeterminantsfromthe

fW eierstrass1.

ants. A s C ay ley remar ed, the idea

array )precedesthatofdeterminant 2. A determin-

dinaprecisewaywith anarrayofn2ordered

wseemsto havebeencleartoC auchywho

eatmentofdeterminants,atreatmentwhich

pontoday3.Unfortunatelytheword deter-

dayto meanbothanarrayanda number

y . (Notetheremar sofMuirin 3, and

fB ocher4. F oraveryclearstatementof the

oremsfromthepresentpointofview,see

br a , d e G ru y te r 1 92 . )

esenttractforan e tendedtreatment

practicallycompletebibliographyisgiven

determinantsinthehistorica lorderofdeve lop-

190 23. Theearlyhistory isattractively

er, V orlesungeniiberdieTheoriederDeter-

03 , p .1 9 . A n e c e ll e nt r e fe r en c e bo o i s

E i n fu h ru n g in d i e De t er m in a nt e nt h eo r ie , e i pz i g

arrayw ithelementsinaf ie ldgandletd(A ) or

nant. A few importanttheoremsare listedforfuture

d(A ).

btainedfromA bytheinterchangeoftworows

d ( A ) .

sortwo columnsofA areidentical,d(A )= 0.

ult,namelyinnotingthatthe hypothesis

henced(A ) = 0, fa ilsw henghasthecharacter-

odifiedtoincludethis case5ortheresult

btainedfromA bymultiplyinganyrowor

t h en d ( B ) = d ( A ) .

s uarearrayeachelementofwhose throw

, t + .. . + d s m (s = i , 2, . .. , n)

sp e c. V o l .3 7 ( 19 2 7) p p . 43 3 4 3 a n d 45 7 4 5 8.

ew . Math. V o l. 50(1855)pp. 282 285.

y techn. V o l. 10(1815)pp. 51 112.

mer. Math. MonthlyV o l. 32(1925)pp. 182 185.

e in e a ng e w. M at h . V o l . 1 7 ( 1 93 2 ) p. 1 9 7.


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dDeterminants.

2 ) + . .. + d ( A m ) ,

obta inedby replacingthee lementsof the throw

n h . i m il a rl y f or c o lu m ns .

sobtainedfromA byaddingtoanyrow(or column)

heotherrows(co lumns), thend(B ) d(A ).

ra y o f n ro w s,

ar , k \

a r,s

ar S t

norarrayofA . A rr " J isaprincipa lm inorarray.

overthe i^ j w aysofse lectingthe numbersi1, . . ., i

,.. .,nwithoutregardto order,andthesign

dingasthesubstitution

.. ., i \

.. r

a rs . e tA r s de no te d ( A \ \ \ ' . . , r, Z \ \ r , \ \ . . . ) ,

a c co r di n g as

, . . . , n

1 , . . ., n

thecofactoro farsinA .

a ip A i t= d p d ( A ) ,

e r s d e lt a .

ndB ben-rowedmatriceswithelementsina

nm-row edminormatri o f theproductA B .

erallselections ofi1,.. .,imfrom1,. ..,n

= d ( A ) d ( B ) .

d. ci. Paris1772.

bra ictheories, p. 49. C hicago192 .


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dDeterminants.

fA isanm. narrayw ithe lementsina

istheorderofaminors uarearrayofA o f

determinantisnotzero1. fA iss uareof

gisca lledthenull ityofA . 2

s uarearrayofordern, withelementsina

g = n , t h er e e i s t e a c tl y l i n e ar l y

nsamongtherows(columns)ofA ,andcon-

ofA sothatthefirst rarelinearlyindependent

mnislinearlydependentuponthese.C all

Thenev idently (A )= (B ). Thereare

n de p en d en t r el a ti o ns

+ h gh e= hh

r + 1 , . .. , n )

B y Theorems7.5and7. 4every (r- - i)-

eterminantwhichcanbe writtenasalinear

ntseachhavingatleasttwoe ualcolumns,

r.

Theorem7. 8

+ . . .+ b p e B % + b pU B f t = 0

thecofactorsB % 1are independentof , sothere

+ b p eC g h + bp hC h h = 0

erforh > , C hh4= 0sinceB iso f ran .

roflinearlyindependentlinearrelationsamong

a st n g , s o r ^ g. H e n c e g = r, n r = .

= 0 , g ( B ) + g ( C ) i n . Fo r e ve r y B t h er e

( B ) + g ( C ) = n .

u a t i on A X = 0 o f r an n g ( A ) i s c al l ed

teby( )thevectoror one-columnarray

e n (y ) = A ( ) c a n be w r it t en f o r

= 1 , 2 ,. .. , n) .

ngew . Math. V o l. 8 (1879)pp. 1 19. Theconcept

plicit , how ever, inapaperby . H eger: Den schr. A k ad.

1 85 8 ) pt . 2 p p. 1 1 2 1.

o p i n s Un i v. C i r c u l ar s V o l . l l ( 18 8 4) p p . 9 1 2

V p p. 1 3 3- 1 45 .


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dDeterminants.

lu t io n o f th e e u a ti o n A B = 0 , a nd i f ( ) i s a n

B ( )isthemostgenera lsolutionof the

ctor(z)rangesoverthesolutionsof thee uation

) r e pr e se n ts e a c tl y ( B ) (A B ) l i n e ar l y in -

n ( A B ) i n de p en d en t s ol u ti o ns o f t he e u a ti o n

e a = n ( B ) s o l ut i on s ( z ) , ( z ) , . .. , ( zM ) o f th e

f thesearee tendedby theso lutions(z( +1)), . . .. (zw )

s o lu t io n s, t h e r a = ( B ) ( A B )

1)), . . . , B (^ r))are independent. Forare lation

) + + C , B ( M) = B ( C + 1 (2 ' + 1 > )+ + C , (i ) )= 0

.. .+ C , = C 1 (/ )+ + C ( iW ),

= C T = 0.

,C arethreematricesofordern withelements

) q ( B ) + q (A B C ) .

by B C . T he n A = > ( B C ) ( A B C ) i s

), (y ), . .. f o rw hichA B C (y )= 0andthe

( y ) , .. . ar e i nd e pe n de n t. T h en ( z ) = C ( y ) ,

a ti s fy t h e e u a ti o n A B ( z ) 0 , a nd t h e ve c to r s

) , B ( z ) = C ( y ) , .. . ar e i nd e pe n de n t. B u t s i n ce t h er e

) ( A B ) s u ch v e ct o rs 1 ( /) , ( z ) , . .. ,

) .

ofthe prdductoftwomatricesisat least

therfactor,andat mostasgreatasthe sumof

1.

n iu s t h eo r em g i ve s ( A B ) ^ q ( B ) w he n

) ^ q ( B ) a nd w he n B = 7 , it g iv es

( A C ) .

w eds uarematri w ithe lementsinadomainof

ield% asasub-variety,thecolumn-nullity

f isthenumberof linearly independentlinear

nsofA withcoefficientsin^ .Thecolumn-

erow-nullity3.

eu B . A k a d . W i s s . 19 1 1 pp . 2 0 2 9 .

w of n u ll i ty . o h ns H o p i n s Un i v. C i r c u l ar s V o l . 3 ( 1 88 4 )

C o ll .W o r s V o l. V p p. 13 3 1 45 .

nn. ofMath. i lV o l. 27(1925)pp. 133 139.

mati . MacDuf fee.


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dDeterminants.

matri ofran ,andd A j ,...A is denoted

m = ( ) , t h en t h e m- r ow ed s u a re m a tr i

. 1

o lumnsofA arrangedsothat >n = j = 0. The

sare linearcombinationsof thef irst row s, so if suitable

first rowsareaddedtoeachof thelast

se lattermaybemadetoconsiste clusive lyof0 s.

theran ofA ortheran ofB .N owbr,= 0

theratio

: bi m

therw ords,

metricmatri ofran thereisatleast onenon-

forder . 2

cof ran . B yTheorem8. 4set

m P i , (t = 1 , 2 , . .. , m)

s n ot 0 . e t b l c p ^ = m . T h en b i f = m pi p .

b eo f ra n < g . H e n ce b k = j = 0 .

ofas ewmatri iseven9.

a nd b r = b sr . e t

m P i = -mi P , p = \ = 0 .

j p 2 t hi s im pl ie s

p p i.

, contrary totheassumptionthattheran o fA

ors.Theorem9.1.A mongtheminorsof

a tri A o forderntheree iststherelation5

. 27A | t

i , m + 2 , . . ., 2 m h = m + i , . .. , 2m

i , h = m + i , h 1 , m, h + i ,. .. , 2m .

ente lements. et ar, = a . Theoperation

ihrungindieDeterminantentheorie, p. 124.

e a ng e w. M at h . V o l . 72 ( 1 87 0 ) pp . 1 52 1 7 5.

e w. M at h . V o l .2 ( 1 82 7 ) pp . 3 47 3 5 7.

ebyG. A . B liss: A nn. o fMath. I V o l. 1 (1914)

. p r eu B . A k a d. W i s s. 1 88 2 I pp . 82 1 8 44 W e r e

3 9 7.


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dDeterminants.

ofcolumnh byelementswiththesamefirst

nd subscriptm,andthenreplacesthe elements

eterminantbyelementswiththesame second

ssubscripth. C onse uently

I h

alto

4 = w i n t he s e co n d su m ma t io n . Fo r h = j , = = 0 .

symmetric. Thecoef f iciento fah , h= \ = , is

mh 8 au /

o fcolumnshand doesnota lterthise pression,

tireleftmemberreducesto1

o ne c e r s i d en t it i es w as g i ve n b y . c h en -

provedthemfrom now nidentitiesina lgebra ic

m e4statedthattheyare impliedinGrass-

u s de h nu n gs l eh r e 1 8 2 , p .

talllinearrelations amongtheminorsofa

nearlydependentuponthoseof ronec er,

dentsystems.H ealsoprovedthatno such

w matrices. Hisresultsaresignif icantincon-

ationsof ronec er sidentit iesbyMuir ,

licatedidentitiesamongtheminorsof

ronec er sinthesymmetriccase. A very

entitiesamongtheminors ofamatri was

. B eaver9.

oc . N a t. A c a d. c i. U . . A . V o l. 12 ( 19 2 ) p p. 3 - 4 .

a th . P hy s i V o l . 32 ( 1 88 7 ) pp . 1 19 1 2 0.

l . A m e r. M at h . o c . I V o l . 2 ( 1 89 ) p p . 13 - 1 38 .

h. A n n V o l .2 ( 18 8 ) p p. 2 09 2 10 .

eangew . Math. V o l. 93(1882)pp. 319 327.

o l . 3 (1 9 02 ) p p. 4 1 0 4 1 .

N a t . A c a d . c i . U. . A . V o l . 12 ( 1 92 ) p p . 39 3 3 9 .

. Math. oc. V o l. 2(1901)pp. 395 403.

m e r . Ma t h. M on t hl y V o l . 3 9 ( 19 3 2) p p . 2 6 2 7 .


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dDeterminants.

ow edthatthereare ustn2 n+ i

norsofamatri . touf fer3showedthat(A J ,

nstitutesuchasystem, thesubscriptsindicatingdeleted

f fer4gaveamethodfore pressingthedeter-

nyorderintermsofnotmorethan14of its

der,andlater5 showedthatthedeterminant

afunctionof fourminorsofordern 1 andone

genera ltheoremonthee pressionofadeterminant

inantsw asprovedbyW. W . F le ner .

m10. 1. fA ise itherageneralmatri o r

,thereisno identity

ai ),

mialsintheelementsofA neitherofwhichis

of A i n de p en d en t a nd d ( A ) = f g . i n ce

ye lement, if auoccursin/ itdoesnotoccuring.

insa11arl, hencegiso fdegree0ineveryarl.

onta insarsarl, giso fdegree0ineveryar, and

lyaslightmodificationisre uiredtoe tend

ymmetricmatri 7.

theoremthat thedeterminantofan

isanirreduciblefunctionofthevariables8.

minantofthegenerals ewmatri ofeven

rationalfunctionof itse lements.

n 2,andtheprooffollowsby induction.

\ i s s e w of o dd o rd er a nd h en ce i ts r an i s n 2 a t mo st .

f b e th e co fa ct or o f ai f in 4 ; ; ; ; ; | | l . A s

5,

tionoftheelementsofA .B yassumption

areofarationalfunctionof thea^ , sothesamemust

beta enas1. B y the aplacedevelopment

a ^ a n = p ta in )2 .

altheoremwasprovedby .Z ylins i9.

ta inedf romA = (ars)by replacingacerta in

s . R o y . o c . o n do n V o l . 1 85 ( 1 89 3 ) pp . 1 11 1 0 .

o l. 38(1894)pp. 537 541.

er . M at h . o c . V o l .2 ( 1 92 4 ) pp . 3 5 - 3 8 .

h. MonthlyV o l. 35(1928)pp. 18 21.

t h. M o nt h ly V o l . 3 9 ( 19 3 2) p p . 1 5 1 6 .

n n . of M a th . I V o l . 2 9 (1 9 27 ) p p. 3 7 3 37 .

t iontohighera lgebra, p. 177. Macmillan1907.

bra ictheories, p. 259. C hicago192 .

B u l l . in t . A c a d. P ol o n. c i . 19 2 1 pp . 1 01 1 0 4.


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27/137

dDeterminants.

of matrices,aswasclearlyshownby

sideredasa linearvectorfunction. . D. B ar-

neousvectorfunctionsofdegreep.H estates

rcterist icroote istsforthegenera lizedmatri .

beenatta inedine tendingtohigherdimensions

thesenseofhypercomple numberinw hich

inceeveryassociativesystemcanberepresented

almatrices,thislac ofsuccessis notsur-

ensionalarrayshavereceivedsomeattentionin

rformsandtensoranalysis.Theirimpor-

eneralizationsofran whichcanbeapplied

utativesystems.Determinantsofma-

uaternionswerediscussedbyH amilton4.

determinantofamatri o f uaternionsby

chtermin theorderoftheir columnindices.

dbi uaternionsandtri uaternionsasmatricesw ith

. tudy7gaveabriefdiscussionofmatrices

belongtoadivisionalgebraor non-com-

derableimportancefromthetheoremof

8thateverysimplea lgebracanberepresented

whoseelementsbelongtoadivisionalgebra9.

overadivisionalgebrain connection

sof lineare uationswereconsideredby

. H ey ting11, and0. O re12. The latterdef ines

ntby

^ O ) + a nA ^ > + . ..+ , ^ / ) ,

tionsofthehomogeneousleft-hand

)

e r . . M at h . V o l . 49 ( 1 92 7 ) pp . 8 9 9 .

a lif orniaPubl. Math. V o l. 1(1920)pp. 321 343.

. Math. Physics, Massachusetts nst. Techno l. V o l. 7

5 . - R i c e, . H . : b i d. 1 9 28 p p . 93 - 9 .

o f uaternions(A ppendi ). ondon1889.

. A m e r .M a th . o c .i l V o l . 5 ( 18 9 9) p p . 33 5 3 3 7.

oc . o nd on M at h. o c. I V o l . 4( 19 0 ) p p. 1 24 1 30 .

at h . V o l .4 2 ( 19 2 0) p p . 1 6 1 .

M . : Pr o c. o n do n Ma t h. o c . I V o l . ( 1 90 8 ) p. 9 9 .

brenundihreZ ahlentheorie , p. 120. Z urich1927.

. : Me s s. M a th . V o l . 55 ( 1 92 ) p p . 14 5 -1 5 2 P r oc .

V o l . 28 ( 1 92 8 ) pp . 3 95 - 42 0 .

A n n . V o l . 98 ( 1 92 7 ) pp . 4 5 4 9 0.

M a th . V o l . 32 ( 1 93 1 ) pp . 4 3 - 47 7 .


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ce uation.

right-e uivalentiftheirrightdeterminants

ricesA andB areright-e uivalentif they

of tw orow sorof tw ocolumns orifA is

iplyingtheelementsofacolumnon theleft,

ontheright, w ith = j = 0 orifA isobta ined

wto anotherroworonecolumnto another

ce uation.

t ion. fA isamatri o fordernovera

, A 2, . . ., A * ' constituten2+ 1setsofn2

arelinearlydependentin ThusA satisfies

11 - 1+ . . . + mf = 0

minimumdegreefi.W eshallcallfi theinde

ca l ar m a tr i i s 1 . v e ry m at r i e c e pt 0 h a s

0,m ).) f ' /.)

) m ( X ) + r ( X ) w h e re r ( ) .) = 0 o r e ls e r ( A ) i s o f

) = m ( A ) = 0 , r ( A ) = 0 . i n ce f i w as m i ni m al ,

nmume uationm(/ .)= 0isuni ue.

nimume uationwillbecalledthe norm

uation. Thematri obtainedf romA = (ars)

thecofactorA , ro fasrisca lledthead o int

a tr i A A j d ( A ) i s c a ll e d th e i nv e rs e o f A , w r it t en

o re m 7 .8 ,

* ( i4 ),

tisfiesane uation

C ^ * - 1 + . .. + C = 0 ,

goniermatriceswithe lementsinaf ie ldthenA

p l)=0 w hosecoef ficientsare in

consideredasan - thordermatri w hose

sin ofdegree^ k . tsad o int

- + D ^ -D -i + . .. +

ngew . Math. V o l. 84(1878)p. 1 3.


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29/137

ce uation.

o fdegree^ k (n 1). Then

p 1 l - l + .. .+ p n

e

p( X ) ,

k - V = Z P n- X " I ,

. h m .

minate.By e uatingcoefficientsofthepowers

a ti o ns a r e ob t ai n ed . f t h e A - t h e u a ti o n is m u lt i pl i ed

eresults added,thesummaybewritten

- 1) -i y ,C k - A *

eoremof H .B .Phillips1whoproved

e - t h or d er m a tr i c es , a nd B . . ., B k a r e ma t ri c es

that

B k = 0 ,

j 1+ . . . + A M w heref t, . .. . & are in-

venis similartoFrobenius 2prooffor

attributestheidea toPasch3.

ymatri satisf iesitscharacterist ice uation

lton-Cayleytheorem,establishedfor

H amilton4, andprovedforn= 2, 3byA . C ay -

minthe generalcasewiththecomment

tounderta eitsproof.Manyproofs,more

beengiven . A . R . F orsyth7applieddifference

. B uchheimmadetheproofessentia lly in

statingthatit wasta eninconceptfrom

81.

e r. . M a th . V o l . 41 ( 1 91 9 ) pp . 2 6 2 7 8.

eu B . A k a d . W i s s . 18 9 p . 0 .

o l. 38(1891)pp. 24 49.

ectureson uaternions, p. 5 6 . Dublin1853.

o s . Tr a ns . R o y . o c . o n do n V o l . 14 8 ( 18 5 8) p p . 17 3 7 .

F o le p ol yt ec hn . V o l. 25 ( 18 7 ) pp . 21 5 2 4 C E u vr es

2 33 .

sMath. V o l. 13(1884)pp. 139 142.

t . V o l . 13 ( 1 88 4 ) pp . 2 6 6 .


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ce uation.

orem14.2, itshouldbenotedthatB uch-

ts o f t he e u a ti o n X I A = 0 a r e ro o ts

e f io fA is^ n.

uationofA forthee uation X I A = 0

eftmemberiscalledthe characteristicfunction

+ t tX * - t n= 0

ationofA , thenttisthesumofa lltheprincipal

l i n X I A = 0 i s t h e su m o f th e d et e r-

a inedbysuppressingn irow sof A

umns.

)= au+ + anniscalledthe trace

ntionofA secondonly inimportanceto tn= d(A ).

uationofdegreenwith coefficientsin% is

t ionofsomematri o fordernw ithe lementsin

onbe

- + . .. + b n= 0 .

ar a ct e ri s ti c e u a ti o n. F or i f i n t he m a tr i X I B

lumnbyX andaddstothe(n lmth, multiplies

m th c o lu mn o f t he n e w ma t ri b y X a n d a dd s t o th e ( n 2 m th

pment, thedeterminantof thismatri isseen

n do n M at h . o c . V o l . 1 ( 18 8 4) p p . 3 8 2 .

s d a n al y se e t d e ph y si u e ma t he m at i u e V o l . ( 1 84 0 )

I V o l .1 1p p. 75 1 33 .

a th . V o l . 21 ( 1 87 ) p p . 17 8 1 9 1. a i sa n t, C . A . :

e V o l . 17 ( 1 88 9 ) pp . 1 04 1 0 7. R a d o s , G. : Ma t h. A n n .

24.


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ce uation.

( o r it s n eg a ti v e) a B e g l e it m at r i " . W e s h a ll

gitthecompanionmatri o f thee uation

educible in% , say

( - * y ,

t, le t

en t t ha t d (X I ) = f ( X ) . T h is m a tr i / w i ll

ri 2.

m inimume uation. Theorem15.1. et

erist ice uationofA , andleth(X )bethegreatest

i ) -r o we d m in o rs o f 7 .1 A . T h en

0

onofA . 3

= C ( X ) . T he n C A ( X ) = h (X ) ( X ) w he re t he e le me nt s

pr im e. i nc e / (A ) = C ( X ) C A ( X ) ,

X )h X ) ( X ),

o ,

) ( X ) .

( A ) 0 a nd m ( X ) g ( X ) w h er e m (X ) = 0 i s t h em i ni -

(f i) = ( X f ) ( X , u ), w eo bt ai n

X I ) -m( A ) = C ( X ) ( X I , A ) .

e nc e

) ( X I , A ) ,

h (X ) g (X ) ( X I , A ) .

ecance led. inceg(X )div ideseverye lement

eelementsof (X ) arere lative lyprime1g(X ) m(X ).

H e i d e lb e rg . A i a d . W i s s. V o l . 5 ( 1 91 8 ) p. 3 - M at h . Z .

5.

es u b st i tu t io n s et d e s li u a ti o ns A l g ^ b r i u e s, i v re 2

s 1 87 0 .

ngew . Math. V o l. 84(1878)pp. 1 6 3.

at h . A n n . V o l . 4 ( 1 90 ) p p . 24 8 -2 3 .


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ce uation.

s t he c o mp a ni o n ma t ri ( o r t h e o r da n m at r i )

0 i s t he m i ni m um e u a ti o n of B ( o r ) .

m inimume uationofB isl i ew ise itschar-

t h e fi r st c o lu mn a n d la s t ro w o f A / i ,

1isobtainedw hosedeterminantisf reeofX X i.

n lmrow edminordeterminantsofX I is

uationof / isl i ew ise itscharacterist ice uation.

tfactorsofthe characteristicfunction/(/)

lein% coincidewiththedistinctirreduciblefactors

(X ).

rem15. 1,

) 0isarooto f / (X )= 0, soevery irreducible

X ). F rom(15. 1)

) .

, /, r,< ui.

m (X ) n .

)= 0isarooto fm(X )= 0, andevery irreducible

X ). 1

erd(A )orn(A )vanishes, bothvanish.

singularornon-singularaccordingasn(A )= 0

ogatory2if itsinde f i islessthann.

non-singularofordern andinde fi,A 1is

r ee f i 1 . 3

ationofA be

1 ' A - 1+ + m ^ X + n (A ) = 0 .

+ m1 A P - 2- h m f, -1 ) n (A ) .

.B utA A 1= A 1A = ,soA B = A A \

\ B = A \

singularoforder nandinde fi,theree ists

ib l e as a p o ly n om i al i n A o f d eg r ee f i 1 s u ch t h at

w, A * - H h m J ) ,A B = B A = n (A ) = 0 .

matri ise ithernon-singularoradiv isoro f zero .

)= 0isthe minimume uationofA ,then

fm(X ) andf (X )haveacommonfactoro f

n: A lgebrenundihreZ ahlentheorie , p. 21. Z urich1927.

c a d. c i ., P a ri s V o l . 9 8 ( 1 88 4 ) pp . 4 71 4 7 5 .

po l yt e ch n . V o l .2 5 ( 18 7 ) p p. 2 1 5 2 4 .

e in e a ng e w. M a th . V o l . 1 27 ( 1 90 4 ) pp . 1 1 1 6 .


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33/137

ce uation.

) ( A ) , f ( A ) = h ( A ) ( A ) , h ( A ) o f d e g re e ^ 1. T h en

/ i a nd ( A ) 4 = 0 . T he n

( A ) ,

en/ (/ ) isa lsosingular.

fm(X )and/ (A )sothat

( X ) f ( X ) .

(^).

) isa lsosingular, andh(A ) isnotaconstant.

,y, ,...)= / rational

)

terminates , y , z, . . . , w herepand arepo ly-

e X , B o f i nd e f i , C o f i n d e v , .. . b e co mm u ta t iv e

, B , C , . . . ) isnon-singular. Thenr(A , B , C , . . . )

canberepresentedasa polynomialofdegree

< f i i n B , o f d eg r ee O i n C , . . .

, C , . . . isbuiltupby theoperationsofad-

ca larmultiplication. inceA , B , C , . . .

ulta teachstepisuni ue. ince (A , B , C , . . .

stsandisapolynomia lin , andtherefore in

e p l = l p , an d r i s un i u e ly d e fi n ed . B y u s i ng

ofA , B , C , . . . , thedegreeof rmaybereduced

ts. etA beamatri o fordernw ithe lements

X I A be itscharacterist icfunction.

f% inw hich/ (A )= 0iscomplete ly reducible .

)= 0arecalledthe characteristicroots1.

,B ,C ,... becommutativematrices,andlet

onalfunction.Thecharacteristicrootsa1,...,anof

. canbesoorderedthatthecharacterist icrootsof

b 1 , cl , .. . ), . .. , /( , b , c n , . .. ) . Th i s or d er -

unction/.2

a s) ,P (o ) = J J ( a> b ) ,y (< w ) = J ( w c ) ,. ..

onsofA , B , C , . . . respective ly. Write

(a (, b , c , .. .) = / ( , b , c , .. .) f (f li ,b ' , c , .. .)

f ( , b , c , ., .) H = ( - ai )f i k . .. + i y b ^ g i i c. ..

k . . . T he re fo re

, c , . .. ) = < x ( ) + ( 3( y) + M y (z ) + . .. ,

,i , = i , . .. , v, . ..

ester.

eu B . A k a d. W i ss . 18 9 1 p p. 0 1 6 1 4 .


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ce uation.

aecommutative , thesetw opo lynomia lsgivee ua l

, y = B , z = C , . .. H e n c e

f (A , B , C , . . . ) . Thereforetheminimum

, . . .) , o fdegreean, ise ua ltoaproductof

ncludeallthe distinctfactorsofthecharacteristic

, B , C , . . . areorderedbyca llingthe i- th

f = 0 /( ( , < , c i, .. .) .

arried throughwhen/isa polynomial

1 , i n y of d e gr e e fi 1 , i n z o fd e gr e e X 1 , .. .

cients,andanorderingoftheroots of

ined. B y Theorem15. every rationa lfunction

obta inedbyspecia lizingthecoef f icientsof

gisthe sameforallrationalfunctions1.

ovedthatthecharacterist icrootsofA nare

hecharacterist icrootsofA . Thespecia lcaseforA

redby . . y lvester3. W. pottisw oode1

1arethereciproca lso f therootsofA . G. F ro -

1 . 1f irstforasinglematri andlaterinthe

errediscovered pottisw oode s7resultand

1878.

mwich9notedthatFrobenius theoremneednot

alfunction.

acteristicrootsofA B arethesameasthose

. 9thecoef f iciento f^A " - inthecharacter-

i s

. - B . p r eu B . A k a d .W i s s . 19 0 2 1 pp . 1 20 1 2 5.

. r e in e a ng e w. M a th . V o l . 30 ( 1 84 ) p p . 38 4

V o l . 12 ( 1 84 7 ) pp . 5 0 6 7 .

o u v . A n n . ma t h. V o l . i l (1 8 52 ) p p. 4 3 9 4 4 0.

. r e in e a ng e w. M at h . V o l . 51 ( 1 85 ) p p . 20 9 2 7 1 an d

eangew . Math. V o l. 84(1878)pp. 1 6 3.

. A c a d. c i ., P ar i s V o l . 9 4 (1 8 82 ) p p. 5 5 5 9 .

M ag . V V o l . 1 ( 1 88 3 ) pp . 2 7 2 9 .

. ' A . : P r o c. C a m b ri d ge P h il o s. o c . V o l . 11 ( 1 90 1 ) pp . 7 5

g .V V o l . 1 ( 1 88 3 ) pp . 2 7 - 2 9 .

b , c , ... ) = 0

c , .. .) .


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ce uation.

nd(A 1, . . . , h i)rangeovera llf j se lections

eset1,2,...,nwithoutregard fororder.

nterchangingthesummationindices.

oremwithoutproof.A proofbasedupon

tyw asgivenbyH. . Thurston1. A proof

T u rn b ul l a nd A . C . A i t e n 2.

atthetheoremofFrobeniusaffordsan

Tschirnhausentransformation. etit

e uationw hoserootsaretherationalfunctiony

. e t B b e t he c o mp a ni o n ma t ri o f f ( ) = 0 .

p(B ) = 0 isthere uirede uation.

bra icintegra lroots, thee lementsofB arerationa l

o lynomialw ithrationa lintegralcoef f icients,


36/137

ce uation.

+ A n t h e a d o i nt - tr a ce o f A 1 , s i nc e i ts s u m wi t h A 1

edef init ionofcon ugatesetby theomis-

andshowedthatforeveryA 1asetof generalized

ists, nota lw aysuni ue ly , how ever. ater2hegave

ndinga llsetsofcon ugatesforagivenmatri .

particularmatri B suchthatofB form

gatematricesinthesenseof Taber,whereto

nity.

tthefield generatedbytherootsofany

phicwiththefield generatedbythematric

acterist icroots. nthisparagraphgis

tA cdenote(ars), obta inedf romA by replacing

ugatecomple number.

ianifH = H C T.5A realhermitianmatri

w-hermitianif = C T. A reals ew-

ew .

U1= U T.7A realunitarymatri is

t or y i f V 1 = V , i . e . , if V 2 = 1 . 8

hastwoofthethree .propertiesinaset,

ary.

al,involutory.

volutory.

utonne9,thelasttwotoH .H ilton10.

atri isuni uelye pressibleA H +

nd iss ew -hermitian.

heorem5.3.

H + whereH ishermitianand iss ew-

arethema imaof theabso luteva luesof the

M a th . I V o l . 2 3 (1 9 23 ) p p. 9 7 1 0 0.

h. Physics, Massachusetts nst. Techno l. V o l. 10(1932)

er. Math. MonthlyV o l. 39(1932)pp. 280 285.

m e r . Ma t h. o c . V o l . 3 ( 1 93 0 ) pp . 2 2 2 4 .

. A c a d. c i ., P ar i s V o l . 41 ( 1 85 5 ) pp . 1 81 1 8 3.

e in e a ng e w. M a th . V o l . 1 22 ( 1 90 0 ) pp . 5 3 7 2 .

d . C i r c. m at . P al e rm o V o l . 1 ( 1 90 2 ) pp . 1 04 1 2 8.

n n . V o l . 13 ( 1 87 8 ) pp . 3 20 3 7 4. P r ym , F. : A b h . G es .

3 8 ( 18 9 2) p p . 1 4 2 .

c. Math. F ranceV ol. 30(1902)pp. 121 134.

eneouslinearsubstitutions. Ox ford1914.


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ce uation.

respectively , andifa= + P isacharacteristic

^ . nh, f3 ^ n .

i s s in g ul a r, t h er e e i s ts a v e ct o r ( ) = ( l , .. . , n )

oa l l 0 su c h th a t (A a ) ( ) = 0 . H e n c e

. ( , / = 1, 2 ,. .. . n)

a j ( } ,

a. i 5,. X j .

acting,

^ ^ i h.

f r i X , .

+ n) s m(V + + * ) ,

n h, f i * n .

racteristicrootsofan hermitianmatri are

= 0 , = 0 , a nd e ve ry / | = 0 .

tsofareal symmetricmatri areall

hy2. Many laterproofshavebeengivenby

uchheimetc. Thee tensiontomatricessuch

adebyH ermite3andresultedinsuchmatrices

racteristicrootsofa s ew-hermitianmatri

.

a = 0 .

ors ewmatricesbyA .C lebsch5and

. H i r sc h : A c t a ma t h. V o l . 25 ( 1 90 1 ) pp . 3 7 3 7 0.

n c ie ns x e r c is es . 18 29 1 83 0 C o l l. W o r s I V o l . 9 pp . 17 4

c a d. c i ., P ar i s V o l . 4 1 (1 8 55 ) p p. 1 8 1 1 8 3.

mericieA lgebrepp. 133 179. Messina1921.

in e a ng e w. M a th . V o l . 2 ( 1 8 3 ) p p. 2 3 2 2 4 5.

. - B . p r eu . A k a d . W i ss . 1 87 9 .


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38/137

ce uation.

thelasttheoremis real,then

# ,) / a r e al l r ea l , th e i ne u a li t y (1 8 .3 ) m ay b e a pp l ie d

1 i ( n- i) v i. -

< 2 3> J k . 1

wasprovedby .B endi son2,andthe

eemstohaveinspiredthew or o fH irsch.

a- -if beacharacteristicrootof A = H + .

emistheleast andMthegreatestofthe char-

hermitianmatri H bedenotedbyhrs.

ueof theratio

e independentcomple variables. Thegreatest

)assumesarecharacterist icrootsofH, for

f + f y , th e c on d it i on s

2, . .. , )

f ^ x ^ = 0.

mvaluewhich/ ( )assumesandMisthema i-

otM. 3

plicite pressionfor and/ f romw hich

smaybederived.

ine ua lit iesbyanothermethod.

arebyA . H irsch: 1. c.

c t a m at h . V o l .2 5 ( 19 0 1) p p . 35 9 3 5 .

son, thecomple casebyHirsch: 1. c.

t h. V o l . 30 ( 1 90 ) p p . 29 5 3 0 4.

n n . V o l . ( 1 90 9 ) pp . 4 88 - 51 0 .

mati . MacDuf fee. 3


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39/137

ce uation.

thatifA isacharacterist icrooto fA andifm

testcharacteristicrootof thehermitian

n m= S X X M .

heelementsofA areallrealandpositive,

whichisreal,positive,simple,and greater

other characteristicroot.

ofthe minimume uationofanhermitian

H hastheminimume uation

) ,

- 1h X ) .

) 2.Thematri H 1= hr,)= m1(H )ishermitian,and

0 . B u t

(H) = 0. B utm1(X )iso f low erdegreethan

mption > 1leadstoacontradiction3.

ofunitarymatrices.Theorem19.1.The

tarymatri areofabsolutevalueunity.

c r oo t o f th e u ni t ar y m at r i U , + i l i s a

r U + U 1 = 1 7 + UC T , a n d i x i s

m it i an m a tr i U U 1 = U U C T b y T he o re m

laries18. 31and 8. 32,

i x = 2is,

tis,

x = r i s,

sgivenbyR . B rauer5. fU= (ur, ) is

1, sothee lementsofUareboundedinabsolute

retrueof thecoefficientsofthecharacteristic

sroots. B utU1andU are li ew iseunitary ,

sof Ularethereciprocalsof thoseofU,

U arethe - thpowersof thoseofU . Unless

U wereofabsolutevalueunity,somepositive

d bemadearbitrarilylarge,thusleadingto

l. A m er . M at h . o c . V o l . 34 ( 1 92 8 ) pp . 3 3 3 8 .

eu B . A k a d . W i s s . 19 0 8 pp . 4 71 4 7 .

M . : A n n . o f Ma t h. I V o l .2 7 ( 19 2 ) p p . 24 5 2 4 8.

uMath. . V o l. 28(1927)p. 281.

u M a th . . V o l . 30 ( 1 92 8 ) p. 7 2 .


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n t eg r al M a tr i c es .

C T = V C T V , a n d if i s a ro ot o f U V \ = 0 ,

w here^ isthegreatestinabso luteva lueand

lueof thecharacterist icrootsofV . ach

ncipa lm inorofaunitarymatri iss lin

ple characteristicrootsofanorthogonal

a lpa irs2.

e uationofanorthogonalmatri hasrea l

rootsoccurincon ugatepairs, w hichby

alpairs.

oremhavebeengivenbyA . . R ahusen3

connectionw iththisla tterpaperseea lsoG. V ita li5.

a t if A + l = 0 h a s r e ci p ro c al r o ot s , A

utory.

aif thecoef ficientse uidistantf romthe

e u at io n of A a re e u al , A + A h - - l = 0

thesecoefficientsoccurin pairswithopposite

A + l = 0 h as a ll r ea l ro ot s.


sina principalidealring.A commutative

ois calledadomainofintegrity.A domain

nt1 inwhicheverypairof elementsnot

mondivisorrepresentablelinearlyinterms

principalidealring9.

, anumberw hichdiv ides1isca lleda

whereuis aunitisreciprocal,and two

darecalledassociates.A setofnumbersof

ssociatedbutsuchthat everynumberof

hemiscalledacompleteset ofnon-associates

egersand 0constituteacompletesetof non-

tionalintegers.

her0noraunit isprimeorcompositeaccording

esnotimply thatborcisaunit.

B r a u e r: T o ho u M a th . . V o l . 3 2 ( 19 2 9) p p . 44 4 9 .

, p ur e s ap p l. V o l . 1 9 ( 18 5 4) p p . 25 3 2 5 .

i s u n di g e O p g av en V o l . 5 (1 8 93 ) p p. 3 9 2 3 9 4.

n . Ma t . t a l V o l . ( 1 9 27 ) p p. 2 5 8 2 0 .

. Ma t . t a l. V o l . 7 ( 1 92 8 ) pp . 1 7 .

o l l . U n. M at . t a l. V o l . ( 1 9 2 8) p p . 5 - 9 .

c c a d. n az . i nc ei , R e nd . V I V o l . 8 (1 92 8) p p. 6 4 6 9 .

u M a th . . V o l . 32 ( 1 92 9 ) pp . 2 7 3 1 .

oerneA lgebraV o l. pp. 39and 0.


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faprincipal idealringisa euclideanring

operties:

ye lementa(e ceptpossibly0)o f thering,

gers(a)called thestathmofa.

ersaandb, b^ 0, theree isttw onumbers

b + r , a nd e i th e r r = 0 o r e l s e s( r ) < s ( b ) .

ple numbersa+ ib,astathmis

mialdomain% ( )o facommutativef ie ld r,

alservesas astathm.Forthetrivialinstance

enas1foreverya. A euclideangreatest

istsineveryeuclideanring.

roperifit isnota fieldandifa stathm

hats(ab)= s(a)s(b)foreveryaandb. ince

)s(b), e ithers(b)= 1foreverybsothatdiv isionisa lw ays

d, ore lses(0)= 0. C onverse ly lets(a)= 0.

reveryb, b= a , and isaga inaf ie ld.

)= s(1)s(b), e ithers(b)= 0(and(5isa f ie ld)or

nit, anduv= i. Thens(u)s(v)= 1 sothat

s(a)= 1, a isaunit, f orthestathmof the

nynumberbyamust be< 1andhencebe 0.

nrings(a)= 0ifandonly ifa= 0,s(u)= i

nds(a)= s(b)ifandonly ifaandbareas-

theseto fa ll - thordermatricesw hoseelements

lring sub ecttotheoperationsofaddit ion

3. 2 { isaringbutnotaprincipa lidea lring.

sca lledunimodularoraunitmatri if andonly

U o f 9 tt s u ch t h at U U = .

matri ifandonlyif d(U)isaunit of

U)d(U )= 1sothatd(U)isaunitof

andd(U)isaunito f ty, thenU1isin D and

definit ion.

A ofW tisa divisorofzeroif andonlyif

matri B o fTheorem15.4(suchthatA B

in 2 R .

adiv isoro f zeronoraunit isca lledprimeif

impliesthate itherB orC isaunit. Matrices

unitsnorprimesarecalled composite.

ositematri canbee pressedasaproductof

primes.

3 . . .,

d e rb u rn : . r e in e a ng e w. M a th . V o l . 1 7 ( 1 93 1 ) pp . 1 29


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atri implies

) d (A 3 ) . .. .

aunit. Thatis, eache lementinthese uence

. d(A 2)d(A 3)d(A 3). . .

receding. uchase uencecanconta inbut

s1.

odularmatrices.Theorem21.1. et

numbersofaprincipa lidea lringw ithgreatestcom-

istsamatri ofdeterminantdnhavinga1,a2,...,an

ueforn = 2. upposeittrueforn 1,

whichhasa1, a i, -1asitsf irstrow , and

g. c. d. d -1of 1, a2, .. . , an - . Determinep

a n = d n. C o n s i de r t he m a tr i D o b t a in e d by

htbya , 0, . . . , 0, 0andthenbelow by

2 A * n -1 . - n - 1 R > - 1. P. Th en d D n) = d n

givenby . Weihrauch4, B ianchi5, and

genera lly thatamatri canbefoundw ith

, an2, annandadeterminant+ 2an2-

sarearbitrary.

eralizedasfollows.A p.narrayofrational

, andtheg. c. d. of thep- row edminordeterminants

betoaddn prow sof integerstothisarrayso

earraysha llbeofdeterminantd. H . . . m ith8

9provedthistheoremanddetermineda llpossible

hortproofby induction.

. Tw omatricesA andB areca lledle f t

saunitmatri UsuchthatA = UB . The

beusedtoe pressthisre lationship, w hichpossesses

e ua lity re lationship, namely10:

d er n e A l g eb r a V o l . p . 4

3, G. isenste in: . re ineangew . Math. V o l. 28

74. F oranyn, C . H ermite : . Math, puresappl. V o l. 14

0 .

o c . Ma t h. F r an c e V o l . 50 ( 1 92 2 ) p. 1 0 0 1 1 0.

M a th . P hy s i V o l .2 1 ( 18 7 ) p p . 13 4 1 3 7.

su l la T eo r ia d e i N u m er i A l g e b ri c i, p p . 1 7 .

m e r. M at h . Mo n th l y V o l . 31 ( 1 92 4 ) pp . 1 1 1 2 .

ng e w. M a th . V o l . 40 ( 1 85 0 ) pp . 2 1 2 7 8.

Ph il os . Tr an s. R o y . o c. o nd on V o l . 15 1 (1 8 1 1 8 2 )

. : A n n . F ac . c i . Un i v. T o ul o us e V o l . 4 (1 8 90 ) p p. 1 1 0 3.

r. Math. oc. V o l. 37(1931)p. 538.


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r A = 5 o r A = f = B .

A .

i m p li e s B = A .

a n d B = C , t h en A = C .

orthere lationA orA = B U.

operationsupontherowsofamatri can

plyingthematri ontheleftbyanelementary

i obtainedbyperformingtheelementary

tionupontheidentitymatri /.

worows.

ee lementsofarow byaunit of

mentsofany row of t imesthecor-

yotherrow,where isin

isaunitmatri w hose inverse isan

esametype. fB isobta inable f romA bya

rytransformations,B = A .1

atri A withelementsin< J 3isthe leftassociate

sabovethemaindiagonal, eachdiagonale lement

mofnon-associates,andeachelementbelowthe

rescribedresiduesystemmodulothediagonal

w illbesa idtobe inH ermite snormalform.

e lementsin ithereverye lementof the

atleastonenon-zeroelementwhichbya

nbe putintothe(n,n)-position. etdnbe

thelastcolumn,andsupposethat

nn = 4 .

e istsaunimodularmatri Uhav ing , b2, .. . , bn

UA = A hasdninthe(n, mposit ion, w here

ra in. B ysubtractingapropermultipleof the lastrow

w s, amatri A 1isobta inedw hose lastco lumn

hemaindiagona l.

th c o lu m n ei t he r e ve r y el e me n t of t h e fi r st n 1

ationof thef irstn 1row sanon-zeroe lement

i , n m p os i ti o n. e t d n- 1 b e a g. c .d . o f

, n -l . n - l ,n -l a nd l et

+ ^ n - l n -l ,n -l = < ^ n - l

e r: M .- B . p re uB . A k a d . W i ss . 18 6 pp . 59 7 1 2.


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1hasO' sabovethemaindiagona linthe last

1, n mposit ion, andeachelementabove

sothatthesecanbemadee ualto0byelementary

essiscontinueduntilamatri isobtained

bovethemaindiagona l.

e inanyprescribedsystemofnon-associates,

multiplyitby aunit.Thisis accomplished

mationofthesecondtype.

leof the( 1mthrow fromthe - th

nbemadeto lie inanyprescribedresiduesystemmodulo

yelementcanbereducedmodulothediagonal

nderstoodthata= bmodO ifandonly if

= j = 0, thenormalformofH ermiteisuni ue.

n= ,buttheprocess isgeneral. uppose

andB

nd = ( lrs)

= 0 , ^ 2 3 = 2 3 a 33 = 0 .

0 , = j = 0 a n d l 1 3 = = 0 . i m il a rl y e ve r y el e me n t

gona lis0.

f^ , Z u, l22, ^ 33area llunitso f5p. Then

dsincea iiandbubelongtothesamesystemofnon-asso-

22 + 3 2

systemmoduloa22,l32= 0.

s = .

non-singularmatri w hosee lementsare

associateofamatri whosediagonalelements

0 f o r r < i , a n d 0 ^ a r i < a u f or r > i . T hi s

aselementsinaeuclideanring ,thereduction

omplishedbyelementarytransformations.

e lementsin . ithereverye lementof the last

ast onenon-zeroelementwithminimum

ninterchangeofthe rowscanbeputinto

fanndoesnotdiv idesomeain, set

s(ann))

in e a ng e w. M at h . V o l .4 1 ( 18 5 1) p p . 19 1 2 1 .


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nsformationofthethirdtype,ain canbe

rchangerowsifnecessarysothat theelement

nisofminimumstathmandproceedas before.

eeveryain.Theproofnow proceedsasin

modularmatri withelementsinaeuclidean

itenumberofelementarymatrices.

yTheorem22. 3theree iste lementary

C / = V ,

aandreduced. inced(U )isaunito f , each

saunito f , andmaybeta entobe1. ince0

temofresiduesmodulo1,U = .Then

.. J E * ,

tarymatri .

unimodularmatri w ithe lementsinaeuclidean

mberofunitsis aproductofa finitenumber

iteset.

lintegersby ronec er1.

modularmatri withelementsinaeuclidean

cesof thetypes


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alintegralelementsandofdeterminant1 is

namely

sors. fthreematriceswithelements

are inthere lationA = C D, thenDisca lled

dA isca lleda le f tmultipleofD. A greatestcom-

)Dof tw omatricesA andB isacommon

multipleofeverycommonright divisorofA

onlef tmultiple(l. c. l. m. )o f tw omatricesA andB

hich isaright divisorofeverycommon

.

rofmatricesA andB withelementsin%

ssible intheform

^

roofo fTheorem22.1, aunimodularmatri U

uchthatthe g.c.d.oftheelementsofthe

nthe(n, n)-posit ioninUF. Thenelementary

eto0everyelementof thiscolumnbelowann.

uedto obtainane uation

sunimodular.Thus

D

ghtdiv isoro fA andB isarightdivisoro fD.

, theree istsamatri = X 1w ithe lementsin


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oc i at e d n t eg r al M a tr i ce s .

rofnon-singularmatricesA andB with

. c. l. m. Muni ueuptoaunit le ftfactor.

nB = 0

fore

ofA andB . fM1isanotherc. l. m. the ir

Ml

tsac. l.m . M2suchthatM= H M2. uppose

.

,-X MB = H L B ,

enon-singular,

= H L .

2 + . ^ 22 ^ 22 = H [ K Y 1 2 Y 2 i , so H i s a ui ut ma tr i ,

fA f1. 1

denoteann- thordermatri w ithe lementsina

F oreverye lementmof theree istmatricesQ

1) A = m Q o r e l s e (2 ) A = m Q + R w h er e

mn ).

he le ftbyaunimodularmatri , itcanbe

rmalform. DetermineQ = ( r, ) , R = (rrs)such

r i ,

s ( m )f or > / , wh il e 0 < s (r if ) ^ s ( m) f or ' = ' .

) = * ( ) ,

r M .. . rn n = s ( r l1 ) s (r 22 ) .. . s( rn ) < s (m ) n = s (m n) ,

andeveryothere lementofR is0, i = m

andsomeelementofR be low themaindiagonal

otheoremsaredueinessenceto . C ahen: Th^orie

aris1914, andintheformherepresentedtoA . C hate let:

is1924.


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ainform(2). etthelastcolumnhavinga

hemaindiagonalbethet-th, andletr ibe

hatcolumn. A ddrow torow i. Then

1 wh er es d (R j ) < s (m n) .

, t he nA = m U Q 1 + U R 1 w he re s d (U R j \ < s m n) .

ndB arematriceswithelementsina proper

d ( B ) = = 0 , t he r e e i s t m a tr i ce s Q a n d C s u ch

a n d ei th er C = 0 o r el se 0 < s d (C ) < s d (B ) .

istmatricesQ andR suchthat

,b = d B ) ,

e 0 < s d (R ) < s ( bn ). f R = 0 , A = Q B a n d

= 0 . f + R = A B A - bQ = A B A - Q B B A

= C B A . T he n

B A ,

s d(C ) s d(B A ) = s i(C ) s^ " - 1 = s d(C ) s(ft) " - 1.

) = s i (B ) .

ereprovedforrationalintegersby .G.du

dedby thismeanstoestablishthee istenceof

. e t bea linearformmodulo forder

ing91. Thatis, consistso fa llnumbersof the

an n .

dependentlyover9 , andthee sare linearly

to R . Thee sareca lledabasiso f .

foranyotherset

> % , ( ' , /= 1 .2 ,. .. , )

odularw ithe lementsin9 isa lsoabasisfor ;

onofthee sisa linearcombinationofthe

onversely,everytwobasesofalinearformmodul

sformation.Thefollowingdiscussionisrelative

. ., enof .

ar f o rm s u bm od u l of o r de r n o f , 2 a nd l e t h a ve

n. Then

' . / = 1. 2, .. ., n)

j schr. naturforsch. Ges. Z urichV o l. 51(190 )pp. 55

lidealring, everysubmodulo f2isa linearformmodul.

eneA lgebraV o l. I p. 121.


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on-singularmatri w ithe lementsin9 . We

G1isassociatedwiththebasisA , , X 2, . . ,

rmatri G1determinesinthisw ayabasis

f 2.

, f it , . . ., f in isa linearformsubmodulo forder

r y nu m be r o f 2 i s i n fi , a n d

g k e .

dwiththebasisfa, fi2,finof 2isC G1,

armatri w ithe lementsin9 . Thisproves

contains 2ifandonly ifG1is arightdivisor

uls x and 2aree ua lif andonly if

C 2 G 2, t h en C 1 C 2 = . s o th a t C 1 a n d C 2

tri G1correspondstothemodulQ1 (G1 iC x ),

sdeterminedonlyupto aunitleftfactor.

rincipalidealringTheset ofnumbers

1and 2constituteamodul ica lledthe

dul(orlogicalproduct)ofthe twomoduls !

sobedef inedasthatsubmodulo f21and 2w hich

ubmodulo f 1and 2.

tainedineither 1or 2orboth,to-

differences,constitutesa modul2mcalled

modul(orlogica lsum)of 1and 2. tmayalso

conta ining 1and 2w hichisconta inedin

1and 2.

1,G2~ 2,Gd^ 2d,Gm~ m.ThenGd

,andGmis thel.c.l.m.ofG1andG2.

of theg. c. r. d. andl. c. l. m . foreachisdetermined

samelatitudeofdefinitionas thatofGd

directlyfromTheorem24.1.

swithrationalintegralelementsto the

gepartduetoA . C hate let1. Hisw or is

s, econssurla th^ oriedesnombres, Paris

nsfinis,Paris1924.

aringw ithunite lement, andlet bea linear

9 whichisalsoa ring,andwhoseelements

seof9 . A ninstanceofsuchasystemisadomain

ociativealgebrainthesenseofDic son2.

,A n n . fi c ol e n or m . l l V o l . 2 8 (1 9 11 ) p p. 1 0 5 2 0 2

ParisV o l. 154(1912)p. 502.

lgebrenundihreZ ahlentheorie , p. 154. Z urich1927.


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nstantsofmultiplicationof beci lc, anddef ine

whichisclosedundermultiplicationon.theleft

a lleda le f t ideal1. im ilarly theremaybedefined

deals.

of is aleftidealif andonlyifits cor-

tisfiestheconditions

1 , 2 , .. ., n )

sw ithe lementsin9 .

, . .. , X n constituteabasisfora le f t idea l% w here

5 i s o f th e f or m

k i gi e ,

iso f theform

ev e ry s l a nd i , t he r e e i s t nu m be r s dr o f 9 s u ch

i c l h eh = d rg rt et .

early independent,

d rg rh .

s t va l ue s d p r o f d r wh e n sl = d l p an d i 8 i .

~ 2 ,d pt rg rh ,

d pr , ). ( P = 1 , 2 , .. . , )

onissuf f icient. etdp randg j benumbers

veconditions.Define

^ k iX iisevidentlya modul. etsbeany number

r er = 2 i ( du, X s

chisthereforealeftideal2.

rrespondencebetweenidealsinanalgebraic

aionalintegra le lements. fX l, X t, . . . , . H

oerneA lgebraV o l. p. 53.

rans. A mer. Math. oc. V o l. 31(1929)pp. 71 90.

co lenorm. llV o l. 28(1911)pp. 105-202.


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i n a n d = l l , X 2 , . . ., X n i s a d ia g on a l

anecessaryandsuf ficientconditionthatA E A 1

artf romadiagonalmatri asafactor, corre-

l.

principa lidealring f tw o lef t idea ls^ i

basesA 1, . . . , X nandf il, . . ., f inrespective ly , theset

isev identlyamodul. ince isaprincipa l

mmodul. t isreadily seentobeclosedunder

yanumberof , so it isa le f t ideal. The

heproductofthe ideals3iand% 2inthatorder.

rrespondencebetweenmatricesand uad-

tanceofthecorrespondenceofthis paragraph,

orrespondingtotheidealproduct thesecond

roductofthematricescorrespondingto the

morphismofidealmultiplicationwith com-

rms.

ngtotheproductofthe twoleftideals31

thematrices

n)

Q2, andX 1, A 2, . .. , X n isabasiso f v

renumbersofana lgebraicf ie ldofordern, aminimal

)isreadilydeterminedfromits associated

g.c. r. d. o f

c ) .3

r ic e s. e t A = P B Q , w h er e e ac h m at r i h a s

dea lring J J . ThenA isamultipleofB .

isamultipleof B ,theg.c.d.dtof thei-rowed

dividestheg.c.d. dlofthei-rowedminordeter-

romTheorem7.9.

withelementsin aree uivalent A = B )

dularmatricesUandV suchthatA = UB V .

c o le p o ly t ec h n. C a h . 4 7 ( 18 8 0) p p . 17 7 2 4 5.

. C . MacDuf fee: B ull. A mer. Math. oc. V o l. 37

a t h. A n n . V o l . 10 5 ( 19 3 1) p p . 6 3 6 6 5 .

e a ng e w. M a th . V o l . 1 14 ( 1 89 4 ) pp . 1 09 1 1 5.


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nceisdeterminative,refle ive,symmetricand

22. )Thepresentchapterhastodow iththoseproperties

iantunderthis relationship.

= B , everyg. c. d. dto f the i- row edminordeter-

tedwitheveryg.c.d.d ofthei-rowedminordeter-

ontherow sofamatri a redefinedasin 22,

bymultiplyingthematri ontheleftby a

ementaryoperationsonthecolumnsaredefined

achbeingaccomplishedbymultiplyingthe

unimodularmatri . The inverseofane lemen-

ntaryoperationofthe sametype.

matri A ofran withelementsin^ ise uiv-

A x , ht, . . ., he , 0, .. . , 0 w hereA i1. 2

alled mith snormalform.

therow sandco lumnscanbeshif tedbyelementary

eminordeterminantoforder intheupper

0. ThenasintheproofofTheorem22. 1, thee lement

anbemade= j = 0 andag. c.d. o f thee lementsof

entsofthe firstcolumnbelowthefirstrow

sbyelementary transformationsontherow s. f

ndsinthe (1, mpositiondivideseveryother

theseothere lementscana llbemadeO' sby

onsonthecolumnssoasnotto disturbthe

f theyarenota lldiv isibleby thise lementa11,

heg.c.d.oftheelementsofthe firstrow,

ewerprime factorsthanan.Theprocess

ementinthe(1, mpositionisobtainedwhich

ntofthefirst rowandeveryotherelement

eeverynumberof% isfactorable intoaf inite

geis reachedinafinite numberofsteps3.

stn 1row sandcolumns, thenw iththe

ndco lumnsetc., A canbereducedtoane uiva lent

, 4 = 0 .

elementofMw erenot0, itcouldbeshif tedinto

posit ion, andA w ouldhaveanon-vanishingminor

i.

P hi l os . M ag . V o l . 1 (1 8 51 ) p p. 1 1 9 1 4 0.

Ph il os . Tr an s. R o y . o c. o nd on V o l . 1 I (1 8 1 1 8 2 )

erneA lgebraV o l. I p. 124.


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umn3, . . . , co lumn toco lumn1, Dis

rem22. 1, there isaunimodularmatri Uw hich,

cesh1bytheg.c. d.ofhl,...,he.Thenew

mentahomogeneouslinearcombinationof

isdivisiblebythe newh1.A gain

m hl, A 2, , A J w herenow h1divides

tlhi+ 1, = 1, 2, . .. , (> 1.

w ithe lementsinaring ' , w ithunit

zero,in whichbothleftandright division

hatis,astathm isdefinedforeverynumber

, andforeverypa iro fnumbersaandb, b4= 0, there

, r, r , s uc h th at

r s( r) < s ( ) ,

o r s( / )< s ( ) .

' iscommutative , it isaeuclideanprincipa lidea lring( .

' i s p ro p er .

mationisoneoffivetypes:

mentsofanyrowofthe productsofany

thecorrespondinge lementsofanotherrow, be ing

mentsofanycolumnoftheproducts of

by thecorrespondinge lementsofanotherco lumn,

tfactor.

owsor oftwocolums.

meunitfactorbeforeeachelementofany row.

meunitfactoraftereachelementof any

yoperationscanbeeffectedby multiplying

ontheleftor ontherightby acertainelementary

lementarymatricesisca lledaunitmatri , in

nceptofdeterminantis notdefinedfor

anon-commutativering.

1hasprovedthatifA hase lementsin ' ,

esPandQ suchthat

, 0 ,. . .. 0 ,

M . : . r e in e a ng e w. M at h . V o l . 1 7 ( 1 93 1 ) pp . 1 29 1 4 1.


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r an o f A , a n d i f ( i s p ro p er , i s b ot h a r ig h t

1

properhadbeengivenessentiallyby . .Dic -

result ingdiagona lmatricesarefactorable intoprime

art fromunitfactors,thesameis truefor

sin@' w hichareof ran n.

ndelementarydivisors. etA beamatri

palidea lring , andlet

,0

lform(Theorem2 . 2). thasbeenseen(Theorem

of the - rowedminordeterminantsofA isas-

o f the - rowedminordeterminantsofD.

di= h1h2...hiofthei-rowedminor

destheg. c. d. di+ 1= h^ h^ . . . hi+1 of the(i+ 1m

sofA .

,h2= d2 dl,h3= d d^ , arecalledthein-

eyare invariantsunderthere lationofe uiv -

duptoa unitfactor.

everye lementne ither0noraunitcanbe

ceptforunitfactors)intoaproductofpow ers

, p h .

ponentsofeachprimefactorformase uence

. .. e u. ( 1= 1 , 2, . .. , )

f asarenotunitsareca lledthee lementary

strass . Theyaredef ineduptoaunitfactor.

resimpleifeach isaprime4.

B ifandonlyif A andB havethesameelementary

.

sameinvariantfactors, theycanbereduced

andhencearee uiva lent. f theyaree ui-

einvariantfactors,fortheseare invariants.

minetheelementarydivisorsuni uely,and

ricesA andB withelementsinacommutative

lentifandonly if theyhavethesameran .

rden: 1. c.

lgebrasandtheirA rithmetics. Univ . o fC hicagoPress

d er n e A l g eb r a V o l . p . 5 .

- B . p r eu B . A k a d . W i s s . 18 7 4.

mati . MacDuf fee. 4


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mentofge cept0isaunit, andanormal

maybechosenw herethenumberof l sis

divisorsisoneof theoldestandmost

anchesofmatrictheory.Theliteratureis so

erisre ferredtoP . Muth sTheorieundA n-

tarteiler1fortheearlypapers.Thetheorywas

rass2forthepolynomialdomainofthecomple

mith3andG. F robenius4formatricesw ithrational

Frobenius5formatriceswithelementsin

latergavearationaltreatmentofthe

edthatifA andB haverationalintegra le lements,

dMthe irl.c. l. m. , thenMA 1andMB 1

meinvariantfactorsasB DlandA D1.

r i . e t A b e a m at r i o f r an w i th

a lringB yTheorem2 . 2, A isaproduct

onalmatri

. .. . 0 .

ducto fmatricesof thetype

actorofhi,and adiagonalmatri

ran . Thematrices 1, . . . , pi,. . . , i have

rethereforeirreducible8.

thefo llow ingdecomposit ionforaunimodular

af ie ldforthecase 4= 0:

onw hen = 0.

edthefactorizationofamatri intoe lementary

- B . p r eu B . A k a d . W i s s . 18 8 p p . 31 0 3 3 8.

P ro c . o n do n Ma t h. o c . V o l .4 ( 1 87 3 ) pp . 2 3 - 2 53 .

eangew . Math. V o l. 8 (1879)pp. 14 208.

r e in e a ng e w. M a th . V o l . 88 ( 1 88 0 ) pp . 9 1 1 .

p r eu B . A k a d . W i ss . 1 89 4 p p. 3 1 4 4 .

. A cad. ci. , ParisV o l. 177(1923)pp. 729-731.

j schr. naturforsch. Ges. Z urichV o l. 51(190 )pp. 55

. -B . p r eu B . A k a d. W i ss . 18 89 p p. 4 79 5 05 .

o u v. A n n . M a th . l l V o l . 1 5 ( 18 9 ) p p . 34 5 -3 5 .


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thateverymatri isaproductofe lementary

i ( ) isobta inedby replacingthe0inrow i

en t it y ma t ri b y . k ( ) i s o bt a in e d by r e pl a ci n g

m n o f t he i d en t it y m at r i b y . " i s t he

rowscycliclypermuted.

everysecondorderintegralmatri isa

nimportantinstanceofaprincipal ideal

main (A )o fa llpo lynomialsin w ithcoef f icients

Theelementsof% (e cept0)aretheunits

) iseuclidean, forifaandb= j = 0aretw onumbers

sttw oothernumbers andrsuchthat

eriso f low erdegree inX than . f is

, thecomple f ie ld), theprimesin^ J (A )are

.

s0 + a rs lX + + a rs l ) w it h el em en ts i n

omia lin ,

o )+ (O r .i )A + + ( ar s ) X 1 ,

triceswithelementsinThematri A

4 = 0 . t i s p r op e r if i t i s of d e gr e e a n d

ndB arematriceswithelementsin^ J (A ),and

, thentheree istmatricesQ andR (Q 1and

R , A = Q 1 B + R l ,

1 = 0 ) o r el s e R ( R j ) i s o f de g re e < l .

.. + A 0 , B = B { X l + . .. + B 0 ,l < k .

0 , t he e u at io n B l X = A k h as a s ol ut io n X = C * - .

k -iX k -lisof degree 1atmost.C ontinueasin

a remainderisobtainedofdegree< Z .

ndB areproperofdegrees and respectively,

er e e i s ts a m at r i Q a n d tw o ma t ri c es R 1 a n d R 2 ,

ectively,suchthat

Q B + R , , P2 = A Q + R 2 , ^ < l ,r 2< k *

hr. Ges. Wiss. Gottingen1909pp. 77 99.

c c a d. n a z. i n ce i , R e n d. V V o l . 23 I ( 1 91 4 ) pp . 2 08 2 1 2.

erneA lgebraV o l. p. 198.

p r eu B . A k a d . W i ss . 1 91 0 p p. 3 1 5 .


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2sothat

P1 = Q ^ B + R 1 ,

ree< / , andR 2is0 orofdegree


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umof thematricesA l, A t, . . . , A k , andw ill

.

h A k .

A ise uivalenttothematri

5n - (X ) + + B . ( X ),

X ) , . . ., h n (X ) i s t he m i th n o rm al f o rm o f X A ,

r w hose invariantfactorsarea ll i sbutthe last

i = = 1 , . .. , 1, f

P n . , Q = Q n + Q n -l + . . .+ Q n - ,

noftherows andcolumnsofPB Q gives

maybechosentobe

V , - 1 H

educible insay

.

n t he f o rm X J , w h er e / i s th e o r da n f or m

eeC oro lla ry15.1. )

yclosedf ie ld , everyht(X ) be ingcomplete ly

rmalformcorrespondingtoeachinvariant

ordanformseachcorrespondingtoanelemen-

A . B yashif t ingof therowsandco lumnsthese

anyorder. ThusB (X ) canbechosen

n ( l) + . .. + n { X ) ,

e ordanformcorrespondingtoanelementary

s e ,: ) .

earetw otypesof invariantforthef ie ld ,

haracterist ice uation, X ^ , . . ., X j , w hichmay

tiona latheoriedessystemslinea ires. Z urich190 .

ationtoA . H urw itz.


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nts,andthee ponentseuoftheelementary

edinvariantsofstructure1.The e ponents

e 1

e22, .. . ) ,. . . constitutethe egrecharacterist ic

fmatrices. Twopairso fmatricesA l, A 2

entsinacommutativefie ld% aresa idtobee ui-

e isttwonon-singularmatricesPandQ

that

P B 2 Q .

ndB 1arenon-singular,thefairsof matrices

a r e e u i va l en t i f an d o nl y i f th e ma t ri c es A l X + A 2

hepo lynomia ldomain (X ) havethesameinvariant

sors)3.

A , B 1 a+ B 2 = B . f A l = P B l Q an dA 2 = P B 2 Q ,

Q foreveryX . nthedomainty (X ), PandQ

determinantsarenon-zeronumbersof

ariantfactorsof A andB coincide.

B havethesameinvariantfactors, there

dQ 1whosedeterminantsarenon-zeronumbers

ay invo lveX , suchthatA = P lB Q 1. ince

heree istbyTheorem29.3tw onon-singular

lementsin% suchthat

1 X + B 2 ) Q

n c e A 1 = P B 1 Q a n d A 2 = P B 2 Q .

erebothA landA 2aresingular, it ismore

mmetriclinearcombinationA lX A 2f iw hose

mialdomain^ (X ,fi)ofhomogeneouspoly-

n^ (X , f i) isisomorphicw ith (a). The

X + A 2f iw il lbecalledthe invariantfactors

epairo fmatricesA ^ , A 2issa idtobeanon-

A 2, u)isnotzero in^ (X , f i) .

= A 1 f + A 2 , A 2 = A 1 r + A 2 s , wh er e f, , r, s

a n d A 2 a r e in a n d if f s r = j = 0 , t h e n th e

+ A 2f iareobtainedf romthoseofA 1u+ A 2v

X + s / n .

n . Fa c . c i . Un i v. T o ul o us e V o l . 2 8 (1 9 14 ) p p. 1 8 4 .

c c ad . n az . i n ce i , Me m . l l V o l . 1 9 ( 18 8 4) p p . 12 7 1 4 .

r eu B . A k a d . W i ss . 1 8 8 p p . 31 0 3 3 8.


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ninverse,definesanautomorphismof

< J J ( , v )b y wh ic h A l X + A 2 f i o o /4 1m + A 2 v .

ngularpairsofmatricesA l,A 2andB 1,B 2

aree uivalentifandonly if theyhavethesame

t, theyhavethesameinvariantfactors. This

heorem30.1.

ingularpair, theree istsanon-singularmatri

. C h o os e r an d s in a ny w ay s o th at p s r = j = 0 ,

+ A 2s. Def ineB / , B 2 cogrediently . Thenby the

e t he s a me i n va r ia n t fa c to r s as B 1 , B 2 , s i nc e A l , A 2

actorsasB l, B 2. nthiscasethepa irsA ^ , A 2

va lentbyTheorem30. 1. Thenthepa irsA uA 2

va l en t .

l isnon-singular, thepa irA l, A 2w ithe lements

th e c an o ni c al p a ir , B w h e re

+ B n- ,

matri o f the i- thinvariantfactoro fA l, A 2.

1isnon-singularthepa irA l, A 2w ithe lements

ie ld ise uivalenttothecanonica lpa ir ,

+ + n- ,

tri o f the i- thinvariantfactoro fA l, A 2.

valenceofsingularpairsofmatrices presents

andP.Muth2treatedsingular pairsof

h3treatedthegenera lcase. . . Dic son4

airsare e uivalentbyrationaltransformations

sameinvariantfactorsandthe sameminimal

fnes. TurnbullandA it en5havean

ngularcase.

rethan onevariableisusuallynotfactorable

eierstrasselementarydivisortheorydoes

applicabletotheproblemof thee uivalence

matrices. . antor genera lizedtheconcept

eometricmethodstohandlethis problem.

uivalenceofpairsofmatriceswas developed

n7. F irstsupposethatl1, .. . , A aretheroots

V o l. 38(1891)pp. 24 49.

o l. 42(1893)pp. 257-272.

nwendungender lementarteiler.Teubner1899.

a ns . A m e r . Ma t h. o c . V o l .2 9 ( 19 2 7) p p . 23 9 2 5 3.

: C anonicalmatrices, C hap. X . Glasgow 1932.

: . -B . B a ye r. A k a d. W i ss . V o l. 98 ( 18 97 ) pp . 3 7 3 81 .

G . D .: T r an s . A m e r. M at h . o c . V o l . 2 ( 1 92 4 ) pp . 4 51 4 7 8.


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\ = 0 a n d ar e a ll d i st i nc t . e t A = ( a r, ) ,B = ( b r l) . T he r e

, . . . , n ), (yn, .. . , y ln), nooneofwhichconsists

= 0 , (i , = i , 2, .. ., n)

0. ( , / = 1 ,2 , )

lizedsothat Z i,iy ibi X i = 1. ince

= h 2 yn bi j k = h 21 yn bu a ,

,

0 , > - < > ^ t = 0.

#re)and = (yrs) areorthogonalrelativetoA andB .

= l l, l2 ,. .. ,X n .

X B \ = 0 a r e no t a ll d i st i nc t , va r io u s

ismultiple , thenumberof linearly independentpo les

e ua ltothemultiplicity . nthiscasethe

ore.Thecasewhenthereare fewerlinearly

emultiplicitypis calledtheirregularcase.

uations

^ ? ( h = 2, .. ., p)

so fX and . fX k isarooto fmultiplicity3.

mberoflinearlyindependentpolesisi, the

X B is

ormations. fPA Q = A , thee lementsof

mmutativefieldg, thematricesPandQ

ctransformationofA withrespecttothe

.

4= 0andMis anarbitrarymatri suchthat

0 t h en

M ) Q = ( A + M )1 (A - M )

nsformationofA .Therearenoothers forwhich

O . 1

Tr a ns . R o y . o c . o n do n V o l . 14 8 ( 18 5 ) p p . 39 4 .


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= ( - A l M) ( + A l M) ,

( A - M ) A 1 ( A + M ).

o feachmemberitfo llow sthat

)1 = ( A + M A { A - M )1 ,

( A + M ( A - M) = A .

e ist, botharenon-singular, andtheydefine

def iningPandQ y ie lds, respective ly ,

, M = A ( - Q )( Q + f .

( P+ ) A ( - Q ),

ree ua l. HenceforeveryPandQ such

( P + ) 1 a nd Q + ) 1 e i s t, t h er e i s an M i n

maybedefinedasin thestatementofthe

andQ + / benon-singularisaserious

asilyavoided.

P A Q = B , d ( Q ) 4 = 0 , t he n f (P ) A f ( Q 1 ) 1 = B ,

ctionsuchthatf(Q 1)isnon-singular.H e

ryand sufficientconditioninorderthatP

nsforminganon-singularmatri intoitselfis

dertheelementarydiv isorsof1 Pand

tcorrespondinge lementarydivisorsareof thesamedegree

aluesof A .

tsinaprincipa lidea lring. fA = PTB P

lementsinaprincipal idealringandif P

congruentw ithB , w rittenA = B . C ongruence

a lence, andisdeterminative, re f le ive , sym-

f . 22. )

x ^^ ofmatri B betransformedbycogre-

matri P intoaformofmatri A , then

epurposeof thew ritertopresentthesub ectas

mthe notionofbilinearform,butthe

yin translatingtheresultsintotheno-

sodesires.ThusTheorem34.1statesthat

ng e w. M a th . V o l . 4 ( 1 87 8 ) pp . 1 3 .


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64/137

sin J 5(Theorem21.1). ThenPTQP isof the

property that 12= j = 0anddiv ideseveryother

yelementarytransformationstheseother

s. Theprocessmaybecontinueduntilamatri

eryothere lementof , oranothercongruent

can beobtainedinwhichtheelementin the

rimefactorsthan 12. B yaddingrow 2,

andthenaddingcolumnssimilarly,a new

tainedwhosefirstrowconsistsof

1 2 ^ 23 , ^ 13 = 2 3 ^ 34 , h n = - 1, n

. ., & -1, n)isag.c. d. o f ( l12, . . ., l ln)andcon-

arto f thisproof , acongruentmatri canbe

butwiththeelementin the(1,2mposition

n-1,n). Unless 12 k i^ l, if o revery i, thisg. c.d.

orsthan 12.

stepsamatri canbereachedin

element.B yproceedingsimilarlywiththelast

o lumns, amatri isobta inedinw hich

g 1 , /,

multipleof row 1torow , 23canbemade0.

vencanbemade0insuccession. Thisprovesthe

issingularifnisodd(Theorem8. ). et

ersA ,,h1,h2,h2,. ..,h^ ,ofthecanonical

antfactorsofQ .

eterminantsare^ h ih ,... h .,where

eroftheset1, 2,...,fi nointegerofwhich

e. tisevidentthenthat theg.c.d.ofthe

antsisdt hlh1h2h2h3. . . to ifactors. Thus

, d d 2 = . h 2, , w h i c h pr o ve s t he t h eo r em 1.

ew canonica lformisuni uee ceptthatthe

byassociates.

einvariants( 27).


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ewmatriceswithelementsinarecongruent

sameinvariantfactors.

os ew matricesw ithe lementsinaree uivalent,

ew matri the2i-thinvariantfactorise ua l

.

d.oftheminorsof thesameevenorderof

er f ec t s u a re 3 .

achotherconstitutea class.

isbut afinitenumberofclassesof non-singular

endeterminant.

primesinaprincipa lidealringisuni ue

ereis butafinitenumberof choicesforeach

mmetricmatricesisby nomeansas

matrices.Thistheoryoccursin theliterature

w ith uadraticforms. There lationA PTB P

isenste in4, w honotedthatifa uadraticform

ormedbyatransformationofmatri P , thenew

tri A .

mmetricmatri ofran withelements

iscongruentw ithamatri o f theform

Q .

ticalwith thatofthefirst partofthe

ni ue,andindeedtheproblemof finding

nonicalformisoneofe tremedif f iculty if it

thasnotbeen attainedevenfortwo-rowed

arerationalintegers,aswillappearin thene t

ntegralelements.Thistopic,which

ortantchapterin thetheoryofnumbers,

ere. C ompletereferencesuptothedate

eninarticlesby . T . V ahlen , and

ngew . Math. V o l. 8 (1879)pp. 14 208.

nius:1.c.

r e in e a ng e w. M at h . V o l .3 5 ( 18 4 7) p p . 11 7 1 3 .

z y l . m at h . W i ss . V o l . 2 C 2 ( 1 9 04 ) p p. 5 8 2 6 3 8 .


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treatisesbyP. B achmann2andDic son3cover

hy .

altheoremwasstatedbyC .H ermite4and

ouff5.

ta finitenumberofclassesofsymmetric

gralelementsofgivennon-zerodeterminant.

)

clusionhere.

ivedef inite . f

a ,

a , t he m at r i A i s c al l ed r e du c ed . v e ry p o si t iv e

oforder2is congruentwithoneandonly

ca lledindef inite. f / isthatrooto f

0

eradical,ands istheotherroot,then A is

s > 1 , f s < 0 . H e r e a ga i n th e re i s a t le a st o n e

s,and usuallymorethanone,butnever

yamethodofGauss thesecanbear-

edforms sothateachchaincorresponds

ttoindicate thegeneralsituation.C an-

finedinvariouswaysso thateveryclass

stonce andatmosta finitenumberoftimes.

uecanonicalformhasbeenattainedon

thoery of matrices mac duffee

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