Adrian albert a introduction to algebraic theories maine press (2007) (1)

polynomialisunique, thatis, twopolynomialsare equalifandonlyiftheir The integern ofany expression 1 of fx iscalled thevirtualdegree of fx has a positive integral degree, or/# = an is a c

>fc

CD

CallNo.5/3 # % 3 AccessionNo^, 33

Author

Thisbookshould he returnedonor before the date lastmarkedbelow,

THE BAKER&TAYLORCOMPANY,NEWYORKJTHECAMBRIDGE UNIVERSITY

KYOTO, FUKUOKA, SENDAIJ THE COMMERCIALPRESS, LIMITED,SHANGHAI

INTRODUCTION TO

By

THEUNIVERSITYOF CHICAGO

CHICAGO ILLINOIS

PRINTED BYTHEUNIVERSITY OF CHICAGO PRESS, CHICAGO, ILLINOIS, U.S.A.

Duringrecent years therehasbeenanever increasinginterest inmodern

con-tents anditsabstract mode of presentation This assurancehasbeen

matterofL E Dickson'sFirstCourse inthe TheoryofEquations However,

Iamfullyawareoftheseriousgapinmodeofthought betweentheintuitive

treatment of algebraic theoryofthe FirstCourse andthe rigorous abstract

treatmentoftheModernHigherAlgebra, as wellas the pedagogicaldifficulty

which isa consequence

at-tempthas resultedinasupposedlylessabstracttreatiseon modernalgebra

whichisabouttoappearasthese pagesarebeingwritten However, Ihave

thefeelingthat neitherof thesecompromisesisdesirable andthatitwould

befarbetter tomakethetransitionfromtheintuitivetothe abstractbythe

mathematics a curriculum which containsatmost two coursesinalgebra

This book is a text for such a course In fact, itsonly prerequisite

ma-terial isaknowledgeofthat partofthe theoryofequations givenasa

chap-ter ofthe ordinary text in college algebraas well asa reasonablycompleteknowledge ofthe theory ofdeterminants Thus, it would actuallybe pos-

sibleforastudentwith adequate mathematicalmaturity, whoseonly

the text in manuscript form in aclass composed of third-and fourth-year

undergraduateandbeginninggraduatestudents,andtheyallseemedtofind

which has been shown me repeatedlybystudents of thesocialsciences

Dr SamPerils during the course ofpublication of thisbook

v

II. RECTANGULAR MATRICESAND ELEMENTARY TRANSFORMATIONS . 19

1. Thematrixofasystemof linear equations 19

7. Rational equivalenceofrectangular matrices 32

III. EQUIVALENCE OF MATRICESAND OF FORMS 36

1. Multiplicationofmatrices 36

3. Productsbydiagonalandscalarmatrices

10. Skewmatricesand skewbilinearforms 52

12. Nonmodularfields 56

IV LINEAR SPACES : 66

1. Linear spaces overafield 66

vii

CHAPTBR PAQBJ

4. The row andcolumnspacesofamatrix 69

5. Theconceptofequivalence 73

6. Linear spaces of finiteorder 75

7. Addition of linearsubspaces 77

9. Linearmappingsandlinear transformations 82

V, POLYNOMIALS WITH MATRIC COEFFICIENTS 89

1. Matriceswith polynomialelements 89

6 Characteristic matriceswithprescribed invariantfactors 105

9. Quadratic extensionsofafield 121

10. Integers ofquadraticfields 124

1. Polynomials in x. There are certain simple algebraic concepts withwhichthe readerisprobablywellacquainted but not perhapsinthe termi-

nologyand formdesirable forthestudyof algebraictheories Weshallthus

We shall speak of the familiar operations of addition, subtraction, andmultiplication as the integral operations. A positive integral power is then

multi-plication

ap-plication of a finite number of integral operationsto x and constants If

g(x) isasecond suchexpressionandit ispossible tocarry out theoperations

two identical expressions, then we shall regardf(x) and g(x) as being the

and g(x) are identically equal and by writing f(x) =g(x). However, weshallusuallysay merelythat/(a;) andg(x) are equal polynomialsandwrite

zero and shall callthis polynomial the ze^opolynomial Thus, in a

discus-sion of polynomialsf(x) = will mean thatf(x) is the zero polynomial.

properties

g(x) =

, + g(x) = g(x)forevery polynomialg(x)

Ourdefinition ofa polynomialincludes the useofthe familiarterm

inde-pendent ofx. Later on in our algebraic study we shall be much more

ex-plicitabout the meaning of thisterm For the present, however, we shall

merely make the unprecise assumption that our constants have the usual

1

aor 6iszero; andifais anonzero constantthen a hasa constant inverse

a""1 such that aar1

1.

Iff(x) isthelabelwe assign toa particularformalexpression ofa

poly-nomialand wereplacex whereveritoccursin f(x) byaconstantc, we

/(c). Suppose now that g(x) is any different formal expression of a nomial in x and thatf(x) =

poly-g(x) in the sense defined above Then it is

g(c). Thus, in particular, if /&(x), q(x), T(X) are

poly-nomials inx such that}(x) = h(x)q(x) + r(x) thenf(c) = h(c)q(c) + r(c)

for any c. For example, we have f(x) = a; 3 2z2

poly-nomialf(x) be carried out, we mayexpressf(x) as a sumofa finite

num-ber of terms of theform axk

maybecombinedintoasingletermwhosecoefficientisthe sumofalltheir

coefficients, and we maythen write

+ . + an-\x + an

unless f(x) is the zero polynomial, we may always take ao^ 0. The

polynomial and we writeg(x) in the correspondingform

with 60 5^ 0, then/(x) andg(x) are equal ifand onlyif m = n, a = 6*for

0,

polynomialisunique, thatis, twopolynomialsare equalifandonlyiftheir

The integern ofany expression (1) of f(x) iscalled thevirtualdegree of

f(x) has a positive integral degree, or/(#) = an is a constant and will be

called a constant polynomial in x. If, then, an ^ we say that the

*

Clearlyanypolynomialofdegreenomaybewritten asanexpressionof theform(1)

of virtual degreeanyintegern^ no. We maythus speakofanysuchnasavirtual gree of/(a;).

de-be done so as to imply thatcertain simpletheorems on polynomials shall

holdwithout exception.

Thecoefficientao in (1) willbecalledthevirtualleadingcoefficientof this

havetheelementaryresults referred toabove,whosealmosttrivial

sum of the degrees off(x) and g(x). The leading coefficient off(x) g(x) is

g(x) are monic, sois f(x) g(x)

con-stantifandonlyifboth factors are constants

f(x)h(x) Theng(x) = h(x).

off(x) andg(x).

EXERCISES*

1. Statethe condition that the degreeof /(x) + g(x) be less thanthe degree of either /(x) or g(x).

posi-tiveleadingcoefficients?

kapositive integer?

4. State a result about the degree and leading coefficient of any polynomial

s(x) =/i +.+/?for t> 1,/ =/(x) a polynomialin xwithreal coefficients.

5. Make a corresponding statement about g(x)s(x) where g(x) hasodd degree

6. Statetherelation betweenthe termof least degreein f(x)g(x) andthoseof leastdegreein /(x) and g(x).

7. Statewhyit is truethatifxisnot afactor of f(x)org(x) thenxisnot a tor of f(x)g(x).

fac-8. UseEx.7 toprove thatifkisapositive integerthen xisafactor of[/(x)]*if

andonlyifxisafactor of/(x).

9. Let/andgbepolynomialsinxsuch that thefollowingequations aresatisfied

other-*Theearly exercises in our setsshould normally be takenup orally. Theauthor's choice of oral exercises willbeindicatedbythelanguage employed.

wise bothfandgarenotzero. Express eachequationin theforma(x) = b(x) and

10. UseEx.8to giveanother proofof (a),(6), and (5)ofEx.9. Hint: Showthat

if/andgarenonzero polynomialsolutions ofthese equationsof least possible

de-grees,thenxdivides/= xfias well as g = xgi. Butthen/iand

g\ are also solutions

acontradiction.

"*

satisfyingthe following equations (identically), thentheyare all zero:

^ 12. Findsolutions ofthe equationsofEx.11 forpolynomials/,g,hwith complex

2. The division algorithm The result of the application of the process

ordinarily called long division to polynomials is a theorem whichwe shall

Theorem1. Letf (x)andg(x) bepolynomialsof respective degreesnandm,g(x) 7* 0. Thenthere exist unique polynomials q(x) andr(x) suchthat r(x)

hasvirtual degreem 1, q(x) is eitherzero orhas degreen m, and

Then, eithern < m and we have (3) with q(x} =

0,r(x) =f(x), ora j* 0,

n > m If Ck isthevirtualleadingcoefficient ofapolynomialh(x) of virtual

k

g(x) is m + k 1.

Thusavirtualdegreeof f(x) b^laQxn'^ng(x)isn 1,andafiniterepetition

of this processyieldsa polynomialr(x) =f(x) b^l

(a xn~ + .)g(x) of

virtual degree m 1, and hence (3) for q(x) of degree n m andleadingcoefficient a^1 ^ 0 If also f(x) = q

Q(x)g(x) + rQ(x) for r (z) of virtual

Lemma 1 states that if t(x) =

sothatg(x) = x chas degreeoneand r = r(x) isnecessarily aconstant,

then r =/(c). Theobvious proofof this result isthe useofthe remarkin

thefifth paragraph ofSection 1 to obtain/(c) = q(c)(c c) +r, /(c) = r

as desired Itisfor thisapplication thatwe madetheremark

a result obtained andused frequently in the study ofpolynomial

definitions and theorems on the roots and correspondingfactorizations of

polynomials* withrealor complex coefficients,to thereader

*

If f(x) is a polynomialin x and c is a constant suchthat/(c) = then weshall

call carootnot onlyof theequation/(x) = butalso ofthepolynomial/(x).

EXERCISES

1. Showbyformaldifferentiationthatif c is a rootof multiplicitym of f(x) =

q(x) then c isaroot of multiplicitym 1 of the derivative /'(x) of /(x).

Whatthen isa necessaryandsufficientconditionthat/(x) havemultipleroots?

2. Letcbe a rootofapolynomial/(x) of degreenandordinaryintegral

for rational

numbers 6, .

, 6 n -i. Hint: Writeh(x) = q(x)f(x) +r(x)andreplacexbyc.

3. Let/(x) = x3+3x2+4inEx.2. Computethe corresponding&< foreachof

3. Polynomial divisibility. Letf(x) and g(x) ^ bepolynomials. Then

by the statement that g(x) divides f (x) we mean that there exists a nomial q(x) such that/(x) = q(x)g(x). Thus, g(x) T divides/(x) if and

poly-onlyif the polynomial r(x) of (3) is the zero polynomial, and we shallsay

in this case thatf(x) hasg(x) as afactor, g(x) isafactor o/f(x).

Weshallcalltwononzeropolynomials/(x)andg(x) associatedpolynomials

h(x)f(x), so that/(x) = q(x)h(x)f(x). ApplyingLemmas 3 and 2, we haveq(x)h(x) = 1, q(x) and h(x) are nonzero constants Thusf(x) andg(x)areassociatedifandonly ifeach isanonzeroconstant multiple ofthe other.

Itisclearthatevery nonzeropolynomialisassociatedwith amonic

coefficient in a conditional equation f(x) = is that used to replace this

equation by the equation g(x) =0 where g(x) is the monic polynomial

associated with/(x)

Two associated monic polynomials are equal We see from this that if

g(x) divides f(x) every polynomial associated with g(x) divides f(x) and

that one possible way to distinguish a memberof the set of allassociates

of g(x) is to assume theassociate to be monic Weshall use this propertyater when we discuss the existence of a unique greatest common divisor

Inourdiscussion oftheg.c.d. ofpolynomialsweshallobtainapropertywhichmay best be described interms ofthe concept of rational function

Itwillthusbedesirable toarrangeourexpositionso as toprecede thestudy

polynomialsandrationalfunctionsof several variables,and weshalldo so.

EXERCISES

1. Let/=f(x) be a polynomialinxanddefinem(f) = xm

f(l/x) forevery tive integerm Showthatm(f)isa polynomialinxof virtualdegreemifandonly

jforeverym > nandthat,if/7* 0,xisnot afactor of/.

4. Letgbeafactor of/. Provethat $isafactor ofm(f)foreverym whichis at leastthe degreeof/.

4. Polynomialsinseveralvariables Someofourresultson polynomials

inxmaybe extendedeasily to polynomialsin several variables We define

a polynomial/ =f(x\,

, xq )inxi, .

finitenumberoftermsof theform

.

We call a the coefficient of the term (4) and define the virtual degree in

Xi, jXqofsuch atermtobek\ + . + kq,thevirtualdegreeofa

par-ticularexpressionof/asasumoftermsoftheform (4) tobethelargest ofthe virtual degreesofitsterms (4). If twotermsof/ have thesameset of

exponents k\,

, kq

, we may combine them by adding their coefficients

k -0,1,

Here the coefficients a^ k are constantsand n/ is the degree of

poly-nomial if and only if all its coefficients are zero If/is a nonzero

nonzero constant polynomials have degree zero. Note now that a nomial mayhaveseveral differentterms ofthesame degreeand that con-

However, someofthe mostimportant simple properties ofpolynomials in

x hold also for polynomials in several xv-, and we shall proceed to theirderivation

polynomial(1)ofdegreen =n^in x =x

qwithits coefficientsa , , an all

polynomials in x\, .

, xq-\ and a not zero. If, similarly, g be given by

(2) with6 notzero,thenavirtualdegreeinxqoifgis ra+n, andavirtual

2, then a and 6 are nonzero nomials in Xi and ao6 ^ by Lemma 2. Then we have proved that the

poly-productfg of two nonzero polynomials /and g inx\, x2 is not zero If we

provesimilarlythat theproductoftwononzeropolynomialsinx\ } . , xq-i

is not zero, we apply the proof above to obtain ao&o^ and hencehave

provedthat theproductfg oftwononzeropolynomialsin#1, , xqisnot

zero Wehave thus completed the proofof

Theorem 2. The product of any two nonzero polynomials in Xi, .

, XQ

is not zero.

We have theimmediate consequence

Theorem 3. Let f, g, h be polynomials in Xi, Xq and f be nonzero,

fg = fh. Theng = h.

poly-nomial or aform in Xi, . , xq if all terms of (5) have the same degree

k = ki + . + kq. Then, if/is given by (5) and we replace x>in (5) by

yx^ wesee thateach powerproductx . x isreplacedbyyk^~ +

*Xj>

f(xij xq)cally in yt Xi, xq if and only if /(xi, xq) is a form of degree k

identi-in xi, .

,xq.

The product of two forms /and g of respectivedegrees n and min the

same Xi, . ,xq is clearly a form ofdegree m + n and, by Theorem 2, is

nonzeroifandonlyif/ andgare nonzero We nowusethisresult toobtainthesecond ofthepropertieswedesire. Itisageneralization ofLemma1.

Observe first that all the terms of the same degree in a nonzero nomial (5) may be grouped together into a form of this degree and then

(6) / =/(*!, ,*,) =/o+- +/n,

where/oisa nonzeroformofthesamedegreenas thepolynomial/and/<is

Ao T* 0. Thusifwe call/othe leadingformof/,we clearlyhave

Theorem 4. Let f and g be polynomials in xi,

, Xq. Then the degree

offgis thesumofthedegrees offandgandtheleadingformoffgis the

prod-uct ofthe leadingformsoffandg.

Theresultaboveisevidentlyfundamentalforthe studyof polynomials

in several variables a study which we shall discuss only briefly in thesepages

opera-tion of division by a nonzero quantity form a set of what are called the

rational operations A rationalfunction of xi, , xq is now defined to be

for polynomials a(xi, xq ) and 6(xi, . ,xq ) 7* 0. The coefficients of

a(xi,

,xq ) and 6(xi, , xq ) arethen calledcoefficientsof/. Letus

, xqwith complex

coefficients has a property which we describe by saying that the set is

closedwithrespecttorational operations Bythiswe meanthateveryrational

duetothedefinitionsa/6 + c/d = (ad + 6c)/6d, (a/6) (c/d) =

(ac)/(6d).Here banddarenecessarilynotzero,and we mayuseTheorem2toobtain

bd 7* 0. Observe, then,that theset ofrationalfunctionssatisfiesthe

prop-ertieswe assumedinSection 1forour constants, thatis,fg = ifandonly

existssuch that//~1 = 1.

polynomials andthe methodof its computation areessentialin thestudy

of what are called Sturm's functions and so are wellknown to the reader

because ofitsimportancefor algebraic theories

Wedefinetheg.c.d. ofpolynomialsf\(x),

, /,(x) notallzerotobeany

monicpolynomial d(x) whichdividesall thefi(x),and issuchthat ifg(x)

di-vides everyfj(x) then g(x) divides d(x). If do(x)isasecondsuch polynomial,

of/i(x), . ,ft (x) isa unique polynomial

of d(x) is at least thatof g(x). Thus the g.c.d. d(x) is a common divisor

of the/i(x) of largest possible degree andis clearlytheunique monic

com-mondivisor of this degree.

If dj(x) istheg.c.d.of/i(x), /,<x) andd (x) istheg.c.d.ofd,-(x) and

//+i(aO, then d (x) is the g.c.d. of/i(x), ,//+i(x). For every common

divisor h(x) of/i(x), ,/,-+i(x) divides/i(x), ,/,-(x), and hence both

dj(x) and //+i(x), h(x) divides d (z). Moreover, d (x) divides/,-+i(x) and

the divisord,-(x) of/i(x), ,/,-(x), dQ (x) divides/i(x), ,fs+i(x).

The result above evidently reduces the problems of the existence and

prob-lem and state theresultweshall prove as

Theorem 5. Let f(x) and g(x) be polynomials not both zero. Then there

existpolynomialsa(x) andb(x) suchthat

isasolution of(10)ifg(x)isgivenby(2) Hence,thereisnolossof generality

if we assume that bothf(x) and g(x) are nonzero and that the degreeof

we put

(11) *.(*)-/(*), hl (x) =

g(x)

wherethe degreeof hz(x)is lessthanthatofh^(x). Thusourdivisionprocess

yieldsa sequenceofequationsoftheform

(16) *,(*) ^ , Ar_i(z) = qr

(x)hr (x)

for r > 1.

Equation (16) implies that (15) may be replaced by hr^(x) =

[q r-i(x)qr (x) + l]h r (x). Thus hr (x) divides both hr~i(x) and hr -i(x) If we

assume that hr (x) divides hi(x) and At_i(x), then (14) implies that hr (x)

both ho(x) =f(x) and hi(x) =

g(x).

Equation(12) implies that hz (x) =

a*(x)f(x} +b^(x)g(x)witha^(x} =

1,

b*(x) = qi(x). Clearly also h\(x) =

a\(x)j(x) +bi(x)g(x) with ai(x) = 0,

bi(x) = 1 If, now, Ai-2() = ai-2()/(x) + 6<-i(a?)g(x) and Ai_i(x) =

divisor d(x) = chr

(x). Thend(x)hastheform(10) for a(x) = car

(x),b(x) =

meansofwhichd(x)maybe computed Noticefinallythatd(x)iscomputed

by arepetition ofthe Division Algorithmon/(#), g(x) and polynomials

se-cured fromf(x) and g(x) as remainders in the application ofthe Division

Theorem 6. The polynomials a(x), b(x), and hence the greatest commondivisor d(x) of Theorem 5 all have coefficients which are rationalfunctions

withrational numbercoefficientsofthe coefficientsoff (x) andg(x).

Wethushave the

COROLLARY. Letthe coefficientsoff (x)andg(x) be rationalnumbers Then

the coefficients oftheirg.c.d. are rationalnumbers

and we shall callf(x) and g(x) relatively prime polynomials. We shall alsoindicate this at timesby saying thatf(x) is prime to g(x) and hence also

thatg(x)isprimeto/(x) When/(x) and g(x) are relatively prime, we use

(10) to obtain polynomialsa(x) and b(x) such that

(17) a(x)f(x) + g(x)b(x) = 1 .

It is interesting to observe that the polynomialsa(x) and b(x) in (17) are

not unique andthatit is possible to define a certainunique pairand thendetermineallothersintermsof thispair. Todothiswefirstprovethe

is prime to g(x) anddivides g(x)h(x) Then f(x) divides h(x)

(17) toobtain[a(x)f(x) +

b(x)g(x)]h(x) = [a(x)h(x) + b(x)q(x)]f(x) = h(x) as desired.

Theorem7. Let f(x) ofdegreen and g(x) of degreem be relatively prime

Thenthere exist unique polynomials a (x) ofdegree atmost m 1 and b (x)

of degree at most n 1 such that a (x)f(x) + b (x)g(x) = 1. Every pair of

(18) a(x) = a (x) + c(x)g(x) , b(x) = b (x)

-c(x)f(x)

For, if a(x) is any solution of (17), we apply Theorem 1 to obtain the

(x)f(x) +

[b(x) + c(x)f(x)]g(x) = 1. We define bQ (x) = b(x) + c(x)f(x) and see that

6 (x)g(x) = ao(x)/(rc) + 1 has degreeat most m + n 1. By Lemma1

-1, a*(x)f(x) + b*(x)g(x) = 1 as desired

If now ai(s) has virtual degree m

a\(x)f(x) + bi(x)g(x) = a$(x)j(x) + bQ(x)g(x)j then /(x) clearly divides

Theorem 8. Letf(x) T and g(x) 7* have respective degreesn and m.

(19) a(x)f(x) + b(x)g(x) =

exist ifandonlyiff(x) andg(x) are notrelatively prime.

For if the g.c.d. of f(x) and g(x) is a nonconstant polynomial d(x), we

have/(x) =fi(x)d(x), g(x) =

0i(z)d(x), 0i(x)/(x) + [-/i(2)ff()l = where

g\(x) has degree lessthan m and/i(x) has degree less than n. Conversely,

let (19) hold If/(x) and g(x) are relatively prime, we have a (x)/(x) +

&o(z)gr(z) = 1, a(x) = a (x)a(x)/(x) + a(x)6 (x)gf(x) = flf(x)[o(x)6 (a;) ao(x)6(x)] But thengr(x) of degree m divides a(x) ^ of degree at most

-m 1 whichisimpossible

EXERCISES

/<(*)

2. Let/i(x), ,/t(x) be all polynomials of the first degree. State their

3. State the results corresponding to those above for polynomials of virtual

degree two

degree of the form (10). Hint: Show that if d(x) is this polynomial thenf(x) =

q(x)d(x) +r(s), r(x)has theform (10) as wellas degreelessthanthatof d(x) and

somustbe zero.

5. Apolynomial/(x) is called rationally irreducible if f(x)hasrational coefficients

andis not theproduct oftwo nonconstant polynomialswith rational coefficients.

Whatare thepossibleg.c.d.'s ofaset of rationally irreducible/(z)ofEx 1?

rationally irreducible, g(x) have rational coefficients. Showthat/(x) either divides g(x) or is primeto g(x). Thus,f(x) isprime to g(x) ifthedegreeof g(x) is lessthanthatof f(x).

7. UseEx 1 ofSection 2 together with theresultsabovetoshowthata

rational-ly irreduciblepolynomial hasnomultipleroots.

possible pairs ofpolynomialsineachcase:

0, there then exists a polynomial h(x) of degree less than that of /(x) andwith

rational coefficientssuch thatg(c)h(c) = 1.

10. Let/(x) be arationally irreduciblequadratic polynomialandcbe acomplex

root of /(x) = 0. Show that every rationalfunctionof c with rational coefficients

isuniquelyexpressible intheform a+bewith aand b rationalnumbers

11. Let/i, ftbe polynomialsinxof virtualdegreenand/i^ 0. UseEx.4

ofSection3toshowthatif d(x) istheg.c.d of/i, ,/*then theg.c.d of /i, ,/<

is 3. Thus,showthat the g.c.d of n(/i), .,n(ft has theformxk

d, for aninteger

7. Forms A polynomial of degree n is frequently spoken of as ann-ic

cubic, quartic,andquinticpolynomialintherespective casesn =

1, 2, 3, 4, 5.

In a similar fashion a polynomial in x\, . xq is called a q-ary

connec-tion with theorems on forms than in the study of arbitrary polynomials

In particular,we shall findthat ourprincipal interest isinn-ary quadratic

are linear in both xi, .

,xm and yi, .

, yn , separately We shall call

such forms bilinearforms They maybeexpressedasforms

andseethat/ maybe regardedasalinearforminyi, . ,yn whose

coeffi-cients arelinear formsinx\, .

, xm.

Abilinearform/iscalledsymmetricif it isunaltered bythe interchange

state-ment clearlyhasmeaning onlyifm = n; and/issymmetric ifand onlyif

thetype ctjx&j fori 7*j. We may writean = a, a;- = a,-i = \c/fori -^j

and have djXiXj = a^XiXj + a^x^x^ so that

We compare thiswith (22) andconclude that a quadraticform maybe

re-garded as the result of replacing the variables j/i,

, yn in a symmetric

bilinear form in x\, .

, z and yi, .

,y by a*, , ,zn , respectively

Laterwe shallobtaina theoryofequivalenceofquadraticforms andshall

yi, - - ,yn) = /(yi,

, yn ; x\, xn ). Thus skewbilinearforms are

formsof thetype

, xn ; xi,

associate (25)only with skewbilinearforms

We shall call (27) a linear combination of xi, . ,xn with coefficients

ai, an Theconceptof linearcombination hasalreadybeen used

with-out the name in several instances Thus any polynomial in # is a linear

combination ofa finitenumber ofnon-negative integral powers ofx with

, xq is a linear combination

ofafinitenumber ofpowerproductsx$ . xj with constant coefficients,the g.c.d.of /(x) andg(x) isalinearcombination (10) of f(x) andg(x)withpolynomialsinx as coefficients.

asecond form,

(28) g = ftiXi+ . + bnxn ,

(29) /+ g = (ax + 61)0:1 + . + (a, + bn)xn

Also if c isanyconstant, we have

The properties just set down are only trivial consequencesof the usual

unimpor-tant They may be formulated abstractly, however, as properties of

coeffi-cients of linearforms) and in thisformulation in ChapterIV willbe veryimportantforallalgebraictheory Thereaderisalreadyfamiliarwiththese

opera-tionsonitsrows and columns

The sequence all of whose elements are zero will be called the zero

property that u + z = uior every sequence u. Evidently if ais the zero

We define the negative u of a sequence u to be the sequence v such

, -an ) ,

se-quencev + ( w). Weevidentlycall this sequence

(38) v u = (61 ai, . , bn an }

for linearcombinationsofsequences are preciselythosewhichholdforthe

forms and that the usual laws of algebra for addition and multiplication

9. Equivalence of forms If/ =f(xi, .

thelinear mapping (39). Ifq = rand the determinant

(40).

is not zero, we shall say that (39) isnonsingular In this case it is easily

, xqand

termi-nology is justified by the fact that the equation f(xi, xq) =

f(xi, ,xq )

, xq ) andg = g(x^ .

, xq )of

intog(y\, . , yq ) bya nonsingularlinearmapping Thestatementsaboveimply thatif/is equivalent to g then g is also equivalent to/. Thus, we

shallusuallysay simplythat/ andg areequivalent Weshallnot studythe

formsdescribedinSection7,andevenofthoseforms only underrestricted

Wehavenowobtainedthebackgroundneededforaclearunderstanding

ofmatrix theoryandshallproceedto itsdevelopment

EXERCISES

1. Thelinearmapping (39) of theform x<= y* fori =

1, q is calledthe

identicalmapping Whatis its effectonanyform/?

2. Applyanonsingularlinearmappingtocarry eachofthe followingformstoan

expression of thetype a\y\+ a^yl Hint: Write /= ai(xi+cx2 + by

com-pletingthe square onthetermin x\andput Xi+cx2 = y\, x2= y*.

4. ApplythelinearmappingsofEx 3tothe followingforms /toobtain

RECTANGULAR MATRICES AND ELEMENTARY

TRANSFORMATIONS

1. The matrix of a system of linear equations The concept of a

studyofthe solution ofa system

ofmlinearequationsinn unknownsy\, . , yn , with constantcoefficients

an ai2

i a22

and is called the coefficientmatrix of the system (1). We shall henceforth

speakofthecoefficientsa/ andfc in (1) asscalarsandshallderiveour

theo-rems with the understanding that they are constants (with respectto the

study of systems of linear equations, but many matrix properties are

natu-ral manipulations on the equations themselves with which the reader is

veryfamiliar Weshall devote this beginning chapteronmatrices to thatstudy

Let usnowrecallsometerminology with whichthereaderisundoubtedly

familiar Theline

19

of coefficientsin theith equationof (1) occursin (2) asits ith horizontal

line. Thus, it is natural to callu* the ithrow of the matrix A Similarly,

the coefficients ofthe unknownsy,-in (1) forma verticalline

(4)

whichwe callthe jthcolumnofA

We may now speak of A as a matrix of m rows and n columns, as an

shallspeakofthescalarsayasthe elementsofA,andtheymaybe regarded

asone by one matrices The notation a^which we adopt forthe element

ofAinitsithrow andjthcolumnwillbe used consistently,and thisusage

willbeofsomeimportanceintheclarity ofourexposition. Toavoid bulky

but shallwrite instead

w-rowed square matrix This, too,is a conceptand terminology which weshalluseveryfrequently

2. Readoffthesecondrow andthethirdcolumnineachofthe matricesofEx.1.

of coefficientsasinEx 1.

2. Submatrices In solving the system (1) by the usual methods the

the unknowns Thecorresponding coefficient matrix has srows and t

col-umns,anditselementslieincertainsoftherowsandtofthecolumnsofA

We call such a matrix an s by t submatrix B of A If s < m and t < n,

the elements in the remaining m s rows and n t columns form an

m s by n t submatrix Cof A and we shall call C the complementary

It willbedesirablefrom timeto timeto regard a matrixas beingmade

upofcertain of itssubmatrices Thus wewrite

(6) A = (A) (t = l, ,s;j=l, ,*),

wherenowthesymbolsAt, themselves represent rectangularmatrices We assumethatforanyfixedithematricesAn,An, .

, Ait allhavethesame

number ofrows, and for fixed k the matricesAU, A2 ;t,

, A,* havethe

same number of columns It is then clearhow each row ofA isa 1 byt

matrixwhose elements are rowsofAH, .

similarly for columns We havethus accomplishedwhat we shall call the

partitioning(6) ofA by what amountstodrawinglinesmentallyparallel to

the rows andcolumns ofA and betweenthem and designating the arrays

of (6) willbe the useof the case where we shallregardA asa two by two

matrix

(A, A

-whose elements AI, A2 , As, A4 are themselves rectangularmatrices Then

AI and A2havethe same numberofrows, A3and A4havethe same

We shallalways mean thattwo matrices are equalif andonlyifthey are

identical,that is, havethe same size andequal corresponding elements

EXERCISES

1. StatehowthecolumnsofA of (7)areconnected with thecolumnsof AI,A2,

A*, andA4

2. Introduce a notationofanarbitrarysix-rowed square matrixAandpartition

Aintoathree-rowed square matrixwhoseelements aretwo-rowedsquarematrices.

Also partitionA into a two-rowedsquare matrixwhose elements arethree-rowedsquarematrices.

2 -1 3 4 5\

1 02-1 -21

01236

4. WhichofthesubmatricesinEx, 3 occurinsomepartitioning ofAasamatrix

ofsubmatrices?

5 Partitionthe following matricessothattheybecomethree-rowed square

notation(6).

a)

6 Partitionthe matricesofEx.5intotwo-rowedsquare matriceswhoseelements

arethree-rowed squarematrices.

7 Partitionthe matricesofEx.5intotheform(7)such thatA\isatwo bythree

matrix;a one by six matrix;a two by twomatrix Readoff A2; A3, andA4and

state their sizes.

thesolution ofthesystem(1). Thereaderwillrecallthatmanyofthe

prop-erties ofdeterminants were only proved as propertiesof the rowsof a

Wecalltheinducedprocess transpositionand define it as follows for

anddefine thematrix

whichweshallcallthe transposeofA Itisan n by mmatrixobtainedfrom

row andjfthcolumnofAoccursinA7

column Notethenthatinaccordancewith ourconventions (8)couldhave

rigidmotionofthematrixwhich weshallnowdescribe IfAisourm by n

matrixand m < n we put q = m and write

wherethematrixA\isagain a#-rowedsquare matrix Thelineofelements

an, a22 , . ,aaqofA andhence ofA\ iscalledthe principaldiagonalofA

is its principal diagonal We shallcall the a,- the diagonal elements of A.Noticenow that A' is obtained from A byusing the diagonal ofA as anaxis for arigid rotation ofA so thateachrowofA becomesa columnofA'

form (6) then A1

is the t by smatrix ofmatrices given by(13) A' = (G/0 (On = A'rfj**

1, ,*;<= 1, , ).

shall passontoastudy of certainfundamentaloperations.

EXERCISES

1. LetAhavetheform (7). Givethe corresponding notationfor A'. Givealso

A'ifAisanymatrixofEx 1 ofSection1.

analogousresult ifA = A',where Aisthematrixwhoseelementsarethe

nega-tives ofthoseofA

3. LetAbe athree-rowed square matrix Findtheform (2)ofAifA = A'and

also ifA = A'.

4. Provethat the determinantof every three-rowed square matrixA with theproperty thatA! = Ais zero.

5. Solve thesystem (1)with matrix

3 for2/1,2/2,2/sintermsoffo,&2,Jt 8. Writetheresults as 2/ == x &A'andthuscom-

pute the matrixB =

(&*/). Dothis also forthesystem(1)withmatrixA' and

com-pare theresults.

4. Elementary transformations The system (1) may be solved by the

method of elimination, and the reader is familiar with the operations on

matrixA ofthe systemand corresponding operationsonthe columnsofA

arecalledelementary transformationson A andwillturnouttobe veryusefultools in thetheory ofmatrices

inter-change of two equations of the defining system. We define this and the

DEFINITION 1. Let i 5^ r and B be the matrix obtainedfrom A by

mul-tiplicationbyascalar) ofsuchsequencesweredefinedinSection 1.8.* The

leftmembers of (1) are linearforms The additionof a scalar multiple of

one equation of (1) to another results in the addition ofa corresponding

corre-spondingresulton therowsofA Thus we make the following

DEFINITION 2 'Let i andrbedistinct integers, c be a scalar, and B be me

by c ofits Tthrow (column). Then Bis saidto beobtainedfrom Abyan

ele-mentary row (column) transformation oftype 2.

Our finaltype oftransformation isinduced by the multiplication ofan

any-wherein our text to results inprevious chapters. Thus, forexample, bySection 4.7,

Theorem4.8,Lemma4.9, equation (4.10)weshallmeanSection7, Theorem8,Lemma9,

equation (10) inChapterIV. However,iftheprefixis omitted, as, forexample, Theorem8,

equation of the system (1) by a nonzero scalar a. The restriction a7*

for or1

. Later we shall discuss matrices whose elements are polynomials

in x and use elementary transformations with polynomial scalars a. We

shallthen evidently require atobeapolynomialwitha polynomialinverseand hence to bea constant not zero In view of this fact we shall phrasethe definition in our present environment so as to be usable in this other

situationand hencestate itas

DEFINITION 3, Letthescalara possessaninverseor1andthe matrixB beobtained asthe result ofthe multiplication ofthe ithrow (column) ofA by a.

Then B issaid tobe obtainedfrom A byan elementary row (column*)

Thefundamental theoremsinthe theoryofmatricesareconnectedwith

thestudy ofthe matrices obtainedfrom a givenmatrix A bythe

applica-tion of a finite sequence of elementary transformations, restricted by the

particular results desired, to A Thus, it is of basic importance to study

the

DEFINITION LetAandRbembynmatricesandletBbe obtainablefromA

bythesuccessive application offinitelymanyarbitraryelementary

transforma-tions. Then weshallsaythatAisrationallyequivalenttoBandindicatethis

by writing A ^ B

see that, if A isrationally equivalent to B and B is rationallyequivalent

Finally, wesee thatifan elementarytransformationcarriesAtoBthere

is rationallyequivalent toB if andonlyif Bis rationallyequivalent toA

*Thereadershouldverify thefact that, ifweapplyanyelementaryrow

sameas thatwhichweobtainbyapplyingfirst thecolumntransformationandthen the

rowtransformation.

Thus we may and shall replace the terminology Ais rationally equivalent

to Binthe definitionabove by AandB are rationallyequivalent.

We have now shownthatinorder to provethat A and B arerationally

samematrix C Asatool in such proofswe then prove the following

A, \ _ /B,

for T by8 rationally equivalent matricesAI andBI and m r byn s tionallyequivalent matricesA2and B2 Then A and Bare rationallyequivalent.

columnsofAinduces acorresponding transformationon A\ and leavesA*

A - <Bl M

A

-\0 A*)'

We similarly follow this sequence of elementary transformations by

ele-mentarytransformations onthelastm rrowsand n s columnsofA

whichcarryA2 to B2and obtain B

Itis importantalso to observe thatwe mayarbitrarilypermutethe rows

ofA bya sequenceofelementaryrowtransformationsoftype2, andlarlywe maypermuteitscolumns For anypermutationresultsfromsome

theoryand shallalsodefinesomeimportant special typesofmatrices We

shall then discuss another result used for the types of proofs mentioned

iscalledat-roweddeterminant ordeterminantof ordert. Itisdefinedasthe

sum ofthe t!terms oftheform

where the sequence of subscripts ii, ,it ranges over all permutations

of1,2, .

, tandthe permutation1*1,

, itmaybecarried into1,2, ,

t byiinterchanges Thatthesign ( 1)*isuniqueis provedin L.E

Dick-son's First Course in the Theory ofEquations, and we shall assume this

re-sult aswell asallthe consequentproperties ofdeterminants derived there.ThedeterminantDwillbe spokenofhere asthedeterminantofthematrix

B and we shall indicate thisbywriting

-|fi|

minorsofA IfA isa squarematrixofn > trows, thecomplementary

sub-matrixofany t-rowed square submatrixBis an (n t)-rowed square

ma-trixwhosedeterminantandthatofBareminorsofAcalledcomplementary

ofsomeofthe most importantresults ondeterminants

Theresultonthe interchangeofrowsandcolumnsofdeterminants

(18)

'

|A'| =

|A|

the computation ofdeterminants, and we shallstatethem nowin the

lan-guagewe have justintroduced

ele-mentarytransformationoftype1 Then |B| = |A|.

ele-mentary transformationoftype2. Then

|B|

=

|A|

.

ele-mentarytransformationof'type3definedforascalara. Then |B| =a |A|.

LEMMA 6. Ifasquare matrix has two equalrows orcolumns, its

determi-nantis zero.

Anotherresult of thistype is

zero.

Finally,we have

otherrows (columns) ofB and Carethe sameas thecorrespondingrows

(col-umns) ofA Then

(19) |C| = |A| + |B|

Thereare, ofcourse,manyotherproperties ofdeterminants,andof these

we shall use only very few Those we shall use are, of course, also well

knownto the reader Of particular importanceis thatresult which might

be used to define determinants by aninduction on order and which does

yieldthe actual processordinarilyused intheexpansionofa determinant

Welet A be ann-rowed square matrix A = (o</) and define AH to be the

complementary minorof aj Then theresult werefer to states thatif wedefine c/* =

( l)i+)'da then

Thus,the determinantofAisobtainable asthesumofthe products ofthe

elements a/ in any row (column) of A by their cofactors c/i, that is, the

"

J

'dt-/.

Theresult (20) is offundamental importance in our theory ofmatrices

LetB bethematrixobtainedfroma square matrixA byreplacing thetth

|B| =0 We expand B as above accordingto the elementsofitsith row

ele-mentsinthe qih row ofA bythe cofactors oftheelementsintheithrow

of A Combining this result with the corresponding property about

Theequations (20)and (21) exhibit certain relationsbetweenthe arbitrary

(i,j = 1, .

, n) .

Theserelationswillhave importantlaterconsequences Wecallthematrix

(22) the adjoint ofA and see thatifA = (an) isann-rowedsquarematrix

which appears in the jth row and ith column of A as the element in its

owntthrow andjth column Clearly, ifA = thenadj A = 0.

EXERCISES

1. Computethe adjointof eachofthematrices

2. Expand the determinants below andverifythe following instances of

spe-cialforms butwhichoccursofrequentlyinthe theoryofmatrices thattheyhave beengivenspecialnames The mostgeneralof these isthetriangular

ele-mentsto theright orall tothe leftofitsdiagonal arezero. Thusa square

matrixA =

(a,-/) istriangularif it istruethateithera,, = forall

j < i. Itisclearthat AistriangularifandonlyifA7

is

triangular; and, moreover, we have

Theorem 1. The determinant ofa triangular matrix is the product ana22

The result above is clearly true if n = 1 so that A =

(an), |A\

= an.

We assume it true for square matrices of order n 1 and complete our

Amatrix A = (a,-/) is called a diagonal matrix if it is a square matrix,

and a/ = for all i ^j. Clearly, a diagonal matrix A is triangular so

thatits determinantisthe product ofitsdiagonal elements

matrix a scalarmatrix and have \A\

=

ajl4 The scalar matrix for which

an = 1 iscalled then-rowedidentity matrixand willusuallybe designated

several identitymatricesof different orders, we shallindicate the orderby

n-rowedidentity matrix.

Anyscalarmatrix maybeindicated by

wherea = anisthe commonvalue ofthe diagonal elementsofthe matrix

Weshall discuss the implicationsof this notationlater,

azero matrix In any discussion ofmatrices we shalluse the notation to

find thatthisusagewill cause neitherdifficultynorconfusion

Weshallfrequentlyfeel itdesirableto consider square matricesof either

oftheforms

Al

A

-A

-whereA!isasquare matrix. Then (24)implies thatA4 isnecessarilysquare,

de-terminants implies that

|

=

|Ai| -|A4|

The property above and that of Theorem 1 are special instances of a

(6) with s = t and the submatrices An all square matrices Then the

|An| . |A|. Evidently Theorem 1

nota-tion which is quite useful LetA be a square matrix partitioned as in (6)and suppose thats = t} theAn areallsquare matrices, and everyA,-/ =