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lll!RODUO TIOB • , • • • • • • • • • • • • • • • •
1
LDJIAR SYstEMS. Y!L"!OllS. AID M.U'iUCIS
)
DS«!R!!f SPAOI S
0~
SY8f'JRS
cr~·
•••••
• • • • • • • • • • • • • • • •
•••••••••••••••••••
OPDAS!IONAL
!!~UA.fiOJrS
IXAKPlil8 • • • • • • • • •
•• •
• •
9
u
• • • • • • • •
14
••
~
•
36
• • • • • •
• • • • • • • • •·
•
.• • •
Moat proil••
equa~lon
ui~
ln applied aa.thematlca lnYolva tol:vtag a.u.
Ol" WTitea ot eqQAti·oDa.
lqu.a.tloaa _,. be
o~
J1J1!LDV ttpaa. bat
the tba.Pl•at u4 •oat bQ.>Onant la the linear equation. If a. aacl
b pP1teaeat lalom qaatit1•• aDd x S.e the u.ab.own, thell u equ­
t1on vh1oh l'educae to the fo'ft u • b 11 called a
b1 one UDlmon.
A'JIT equat)1on in oue
to th1a foa ia .aid to b•
thaA llae&l' tnea.
Some
UJllmo1m whiCh 1.• not l'lduct.lla
~n-U.aeu.
great D1"1at7 of fol"'la, .a.n.d
~
11~ ~tlon
lon-lhea~
equattona have
&
genel"all.T lii10h ao" cUtf1GUlt to aol••
«JAJ~Plee
of non-U.neu- f1Cl\1&t1ona
~c
(l) ~ - ):r • 0 •
(2)
(3)
ain
c. + • -4a • 0 •
.Ji + 5 •
0 •
lqu.tloa (1) baa thl'.. aoluUona. (2) baa t.nfb.dteq JllllV' aolu­
tlona, aDd (3) • • no Jel.u.t1ona, 1t ve
real
uuai•~•~
ar.
aeaklng a l"oot
41110IIg the
The existence of a aolmtioa for a given eqwation 4ePtDda
to a large extent on the razage ot 'faluta which we all.ov the UllknoVD to
.baTe.
l'o!! uuapl$, U'
x
io
tion }x • 5 at no aolut.iQJl•
tor z, then
lf a
;t.;equl.~ted
lt, howne, we penait fltaotloaal T&laaa
th1t tqU&tion haa the
aDd b are
~tlob.al
t .o be an lntegd-1 theh the eqUA­
~ot
x •
J•
uaaba?a• then ax • '
Et l8 w14ot that
&awav• baa a
lolution
2
which is a rational. number. provided that
=0,
no solution uiets unless b
f1es the equation.
a
is not zero.
and then every value o:f x
In thia ease, we 8aJ that
x
eat1e­
is a.:rbiira:a.
is not the aa.me si tua.tion a.s in e.mmple (2) above.
= 0,
If a
Thia
Equation (2) has
1nf'1niteq ma.ny eolutions 1 but these are a aeries of epeeif1c mlllbel!'e:
(2) 1a
.at.1 ea.tisfied
by all values of
s •
The baste number eystem which will be employed throaghout thie
paper is the complex number field.
the tt.Pe
a + bl, where
a
&nd.
A Untar equation ax
= b,
baa the unique solution x
*
b
a•
b
A CoJIIPlex llUlllber is a.rrt ntlllbel" of
al!'e r-eal numbers axd
where a
and b
unleaa a •
o.
x
o
~
is arb1traJ7.
As before.. if a •
o,
If this CoDd1•
Note that the solution
ts generally a. complex UllDiber.
The cpmplg coGwm:to of a+ bi
.J=i .
a.re complex llWilbere,
then b mat be zero in ol"der that a solution exist.
t1on is aat1ef1ed, then x
1 •
te the number a - bi •
3
Only the simplest mathematical problems lead to a single e~tion
in one unknown.
More complex problema Will require the colution of a
If all the equat ions a.re linear
set of equations in sever-al unknowns.
in ea.eh of the unknowns, this set is called a linEar system of equa­
tions.
Sa.ch a qste.m mtq' be wr1 tten in the following fol't!l:
(2.1)
• • •
TJ'I..ie 1e a S7atem of m
• • •
e~tions
• • • • • • •
n unknowns.
in
;,• ••• • ~, and the known quanti ti ea are the
a 1j
~e unknowne
and the
are
71 •
!he
systeJil (2.1) can be expreaud in a mu.oh shorter f'ol"'m b1' using the
~tion
notation:
n
t
.1-1
a 1SXJ
• 71
,
(1
= 1,
••• , m) •
This e;zpression mea.ne precisely the same thing as (2.1).
ao~utioA
m
of (2.2) is 8Zf3' set of values tor ;,, ... •
eqUation• are satisfied simu.ltaneouall'.
are complex
1
complex llWilbers.
and
y-
n"Wnbef's~
*b.
A
Btlch that all
Ve shall su.ppose that
a 1J
and that the eolut1on must be a set of
Another Ya7 of dealing with linear qatems is to use the concepts
of vectors and matrices. We shall ti'eat these topics by
following definitions .
Defip.itio:q. 2r:l
(2.
the
pp .l~)
An ;n=dimep.aiop} complg yeq3;.ot is
ot n complex numbers,
maJ.d.Jlg
w~itten (~ ••••
M
ordered aet
• ~) ~ x • 'fu& totality- of
such Yectora tor- a given n. is called an n-dimeneiona.l eomple.x·
vector apace.
The nwnber z
is called the 1-th component of vectol" x •
1
The zero vector 0 21\1 (O, ••• , 0) is the vector all ot whose com­
ponentt are eeroe.
EerG
Note that the symbol 0
is used to represent the
vector as well as the complex l'l1llllbe1' eero.
~f!:g,&Uon 2.2
The .mam of
tWfi
vectors x
=("1,,.
... •
~)
and
is the veeto:t x + 1 • (;_ + r 1 • ••• • 2n + Tn) •
ot a. vector x 07 a eompl.e x nwilbel' a is the vector
7 w (71 , ••• , 711 )
The
a:Jt' •
vqjy,s»
xa • (~ • •• • • a.xn).
Two vectors
x and
y
are tml!i if and on.l¥ if x ... 7
= o.
A. vector apace with the above p~perties ts called a },(inee;r
veetora x and y
IPM,••
is the complex nuiber
-
where 7s, 1e the complex conJugate ot 11 •
The equations (2.2) are ea14 to define a lineM> homogeneous tl'a.nJJ..
formation of the n-dimensional vector x into the m-dimensional
veeter 1' ,
.
Defi~tj.on 2,J;
is the
:rectaJ.~gU.~
!fhe W;i& of the transfol"ma.tion defined by (2,2)
.e:Jr.ta'3'
•
If m • n,
is called a sgwe
.A
etrix of
order n •
We can now Write the aystem (2,2) 1n the foi"m
~s
qmbolte notation meatus that
which t.ranef'orms x
equation ax • b
l!tttn!ttga
into '3'•
A
is thought of as something
!he sim1lar1t;r betWe$Il {2 •.3) and the
is evident.
~·5
!fhe Jfill of two matrices A
ie the matr1% A+ B
= (&1.1 + b13 )
= (a.1j) and
J3
= (blJ)
which t~anaforms tne't'f x
into
.Ax+Bx.
The m:®U$ of a mtrb:t by a complex nwnber a. is the matrix
aA
= (cr.a1J)
which transforms eve1'7
x
into
In the rest of this chapter ve shall
Go( Ax) •
as~e
that m
=n
•
In the
linear ay.atelll (2.2) • we can al.WaJ'I su.ppoae m • n by d.ef'iniag a1.1 -= 0
and
r1 •
0
tor 1
n~tmensional
>m•
Tim• we &ball be concerned only with
•ectors and n-th order square matrices.
6
for
i.J
=l,
••• , n • The
te~o mat~1x
transforms every vector x
into the tero vector.
The 3mU maj;ri% is
for
i
r/: J • fhe
I = (& 1 J)' vhere
unit matrix carries
x
&11 • l,
and
into 1tseli':
8ij
Ix
=0
=x
for
every x •
D,!fip.&.tion 2.]
a vector.
fhb proflu.et
o·f a matrix A and a vector x 1a
In one case, we have
(i • l, ••• , n) •
•
where the quantity in parentheses is the 1-th component of the vector
Ax •
This defines pt"eciselr what is meant bf the tranoforme.tion
"We may also have a product with the vector on the lett:
xA •
Genel!&lq,
xA
.Detf.n1t1Qn 2~§
I
~.~
\t=l 1 1J'\J/ ,
x a
(J • l, ••• , n) •
Ax •
The DtQd.ugt of matrix A
= (a1 j)
by matrix
B • (b 13 ) is the matrix
u
~~1 •n~J
which transforms every x
Dt(lA1~1oQ
2e9
into A(k) •
If a unique matrix B
exists such that
1
All • :BA. • I •
is called the invtt'l of A and is written A-l •
.B
Detini~iQP, ~.lQ The ma.tri:.r:
of A. (a1Jh
A.
(aij)
is the
A'
0
(aJ 1 )
is called the tran&Pi'!
of A;
COJl,.1ggatt
and
r =i' ie
the
Hermitiy. gopJyate of A •
We are now prepared to diseu.es in some detail the solution of the
equation A:% • 7•
dependi~
Here there are three eases,
on the
character of matrix A • First , 1! A has an. inverse A-l• then we
have
•
or
z • A-ly •
This gives the uniqu.e solution x which Mti$fiea the equation.
Second. if A •
vector
o,
y = o,
there. is no tolution unless
x is a solution.
~hese
cases are like those for the simple
Bu.t. tor Ax • 7 • there is a third case, since a
equation ax • b •
matrix 1JJB.'Y not be zero and yet haTe no inverse.
to be s1Dgal.ar.
onlY if y
If A is a. singular matrix,
aatisfies certain restrictions.
some of the x1 are a.rbitra.ry.
completely. (2, pp .6-7) .
mopiM
(z,
y)
2,~
= 0 tor
and. then evef7
fying the equation
zA
=o
s
a. solution will exist
If these
~e
satisfied,
The tollorillg theorem 82;plains thi il
The syetem Ax • y
all vectors
atoh a matrix is said
has a solution if and only it
~ttch that
sA
;:J
0 • A.;ey
z satis­
can be witten as a linear combination ot
8
z' a; tor emmple,
certain linearly independent
d ~ n •
l
B ' ••• • I
•
Iiov 7 m.st sat1ef'7 the d linear homogeneous equations
(zk, 7) • o, k = l, ••• , d •
E:xaotl.y
d
of the components ot x
can be chosen arb1tra.ril)-, and the remaining n - d
detel'mined.
and
d
n .. d
fh.e integer d
1 s the
~
will then be
1a oaJ.led the yfeet of system Ax • 1' ,
•
fhe proof o-f this theorea wlll not be g1ven. since it follows
veey eloeeq the proof of theorea
involving inverses.
5.1, without the restrictio.n
9
In definition 2.1. the idea of an
wa.a introduced.
~imensional
vector space
Now we shall extend t he concept of •apace".
:First,
we may consider veetQra with a denumerable infinity of components;
the totality of such vectors is an 1nt1n1te-dimensional vector
space.
A
further extension might be to vectors w1 th a non-denumer­
able 1Dfin1ty of components.
of a
cont~ous
variable
Ve can treat such veetors as tnnct1ona
a. !he totality of
c~rtain interval a ~ a ~ b
~otions
1• called a gunction
In general, an abstract linear space
defined on a
snace•
S consists of a set of
elements which have t he following properties:
(l)
It
x and y aze elaenta of s, the
defined and is in
and comma.tative.
x +0
that
(2)
If
x
a
x
S •
ih1a operation is associative
A zero element 0 e:xists in S 8\lch
tor all x
in S •
~
is defined and u
operation has the property that
a. • 0
or x • 0 • or both •
If x
and y
if
x + 1 ia
is in S and a. is 8.1'1¥ complex IIWnbe1t, the
product ca •
(3)
SWI'l
belong to
is in S • 1'hia
<a= 0
S , then x = y
if alld onl.7 i:t
if aJJd onlf
X - "¥ a 0 •
Another operat ion is often def ined tor abstract spaces.
operation is called the inner product
!hie
(x, y), which is a complex
10
~he
l'lWilbel!' aasoeia.ted with x and 7 •
inner product must oa.tis:t)'
the relation
(x,
(3.~)
where (y, x)
y) ~
(y, x) ,
denotes the comple'x conjugate of (y. z) •
!he ieer product for vecto:r epa.ees was given in definition
2.3 • For function epaCeJs, the illl'ler product is
=Jr
b
(3.2)
(x, y)
x(a) y(s) ds •
a
Here it iB assumed tba.t
a aiagl.e :real variable
to
x(s)
o ,
and y(a)
are complex tu.nctions ot
and. all elements
x and y
belonging
are integrable on the interTal. a ~ e ~ b •
S
tn abstract spaces, one utu.alq d.oee not define a product in
the o'i'dinaq aen&e.
which a product
Tbat is. we do not consider
-q is a.a element of the spe.C)ft.
nm.l.~ipliea.tion
In the next aec­
tion, we ahe.ll, howev-er. consider transformations in abstract
spaces.
in.
u
A transformation in an abstract spaoe
element
x
1n
y
S a.nothel' element
in
~elates
S
s.
to each
Su.oh a transformation
may be written tn the following notation:
(4.1)
(x. y in S) •
!rhe a,ymbol
tra.nsfol"mation.
At called an
We shall su.ppoee
that 1a, tor ea.eh x
*
'Phe
in S
1a used to rep.tteeen.t the
that A is a single-valued operatott;
ta a unique element of S •
h
Jltf1N.t1on l}.,J. fhe
eveey- x
o-!erato~,
.am QJ.letttar:
O 1 s the operator which catrles
into the zero element of
s.
0%
=0
for every x •
iiMtiy QRmtgr I caniea eveey x into i tselt.
Ix • z
tor all x in S •
PefiJ1U1oa ~,2 .An operator L
is said to ·ne a4d1t1t.t 1f
L(x + y) ~ Lx + ~ tor i l l :r. and 1
which is continu.otu is a
A!Detu:
1n
Pl?ttato;:.
S •
An addi t ive operator
The notio.n ot continuity
involves topological eoneepte which do not eoneern us here.
lf·e shal l
hereafter use the telfm linea.:t* operator, although we ba.ve not ade­
qu.a.tel.7 defined it,
Dtfiaition
4,1
~he
euro of two
ope~tore
operator A + B WhiCh transforms :z: into
of two operators A and B
A and B ia the
Ax + llx •
!b;e proQllQt
ia the operator AB whieh can1ee x
12
into
A(lb:) •
It can readily be shown that addition of oper-ators is aasooiat1ve
and
commt1tative~
plication is
that mu.ltipl1catlon is associative; and tba.t mu.lti•
distributi~e
over addition.
Def1n,jlon 4.4 If a unique operator
B exists ~~
that
AB • BA. • 1 • it is ea.lled the invel"ie of A a.nd is written B
= A•l.
J'or a linear opera.tor L , the invnse, lt it exists, will be
lt is asSWDed that the 1nTersee of all U.n-.r opera­
d.enoted by M •
tors considered here ue s1Dgle • valued and linear.
De1'1nit,12A
is denoted by
4.5 The Rumttiap. gon.1m;ate of a linear operator L
V
and is defined by the l'ela.tion (L'#x, 1') • (x, Iq) •
which mat be eati sfied for all x and 1
in
S ,
In a Tector apace, linear Qperatore are matrices.
For function
epac•a, operators are represented by integrals:
f
.b
(4.2)
I.(a, t) x(t) dt ";r(a),
(a~
1
~b)
a
!he Hermitian conjUgate of an integral operator ia
J
b
11
J. x"
I. (t, a) x(t) dt ,
a
The operation
xL
may be defined
(&
~ • ~ I>)
•
,
13
:iL•
f
b
x(t) L(t, e) dt,
In abstract epac:e., we mq consider
(o
<
c b)
a
•
to be a linear
Lx • '3'
equation in Vh.1ch L and .,- are known and x
L
(a
If
is to be found.
bas an inverse M , the equation ha.a the uniqu..e solution x • M7 •
li' · L • 0 • the equation has no· solution unless
eveey x
in S ts a. solution.
stUl he.Te DO inverse.
y
=0
., and then
However • .1. might not be seo and
It seems ltkeq that !heol:!em 2.1 would
here as 1t does for vector epa.c.ea.
a~
Aa atated for abstract apace••· it
is written in the followtng form:
'£!l!iiQlj.lf
2.1•
The equation Lx • 7
(s. 7) • 0 tor all s
su.oh the.t
1L
baa a solution 1f and oal7 it
=o •
A theor81l' aimila.r to this baa been proved for i:ltegral. equations
of Fredholm t;ype ami secolld kind (1. pp.l01-1oq).
onl¥ those llne&l'
opemto~a
We shall conaicler
vh1oh aa.U&f7 this theol!'em •.
SYSm.tS OF OPERAT IOIAL E(tT1M'IONS
Now we shall be concerned with problems which involve more than
one apace.
equation
s1
11'1rst, su.ppose we haTe two spaces
L;,
=~ (;_
in
~ •
22
s2 )
1n
mation which carries each element of
s1
sa.y
~
~
that operator L
$pace
s1
!he detiniti.ona of' section 4 appq
s2
and
~
Then the
indicates a transfor­
s2
into a.n el&ment of
apace
s2
W•
•
•
in an obViO\ls way to the
present situation, and will not be restated- Note that the identit,­
operatoll'
X
elw~e
maps a epace into itself.
1t is an operator which mapa spa.()e
M ,
L1]_
the eolut1on of
=~
s2
It L bas an inverse
Let LJi be a
If
lin~
•
In this case,
ie ;, • M ~ •
Su.ppoee we have two aets of linear space&:
t 1 • • • • , I• • . m ~ n •
s1
into
Oona1der elements
x1
~-• ... ,
in
Xs,
lb
and.
and
YJ 1n
rJ •
operator which maps apace X1 into space TJ •
tJi ha.D an inverse._ t .t is written MJl and is a. ltnea:t opettator
which maps
YJ
into
x1 • ll:e contider the linear 81"Btem. of opera­
tional equations
~hi•
qstem is aa.1d to have a solution for given Ljl and 7 J it
set of elements
li.J.• ••. ,
~
are satiatie!ld s1mltaneousq.
&
exists w.ch that all m equation•
By defining
31 = 0 and 7J • 0 tor·
L
J > • , ve TlliJ:¥ su.ppose m • n • It 1s asawned that th1s has been done.
J'or convenience, we sha.ll d.e t1ne vectors and matrices fo"tJ
15
operational
e~tiona.
=
The vectgr x • {x1 } {:1)_ 9 . . . • ~J is an
n-dbaenaional vector whose 1-th component is an element of apace
Definition
5.1
Det1n1.t iqn
5.2
whose
!the ..uzg, ttctQ£ ie
0 = {0} ::: {o, ... , 0} •
1-th component is the sel'o element of the
The inner product
of the tTPe
x17 1
Dtf1nit1on
(x, 7)
1-th space.
will not be defined because product 1
are undefined.
513 The
matriJ L e (Lji)
nents are linear operators.
is a matrix whose compo­
Hereafter, the
~bol
L without anb­
aor1pte will be used to denote this matrix.
Def1n1 t ion 5 .)t. rhe ptoduct (Jf matrix L a.nd veeto:tt x
is the
:following vector:
Th1e is a
veeto~
whose
j•th component 1a an element of space YJ •
lilv1dentl3', qstem (5,1) 1s equ1nlent to the equation Lz • 1 •
Dtf1ait1on 5.~ The· yector OJorator
zP
= {zP}
1
• {zP
1
, , •, • z:} ,
where
zP
1a
~ 1s an operator which m&ps
16
D!fa,p,itlcm. 5•6 The prgduct
of vector operator zP
o.n.d matrix
L 1a the vector
Zp L •
{~"
zP
j
J-1
x,
·
}
ji
whose 1-th component is an operator which maps sp.aee
x1
Dttinitipn 5,7 !he inner Rrodu.ct of vector operator
v ector y
7j
zP with
is theelement of apace YP defined by
( zP •
where
into YP •
zP
T> =
is the oomp1ez eon~ate of
It seema as if Theorem 2.1
t
~
yJ
•
lhould be valid for w,retem (5.1) •
Unfortunate]$, the attempt to prove this has not been su.cceasft11.
A
more specialised theorem will be proved in this section, vh:lle sec­
tiou 6 is deToted to a. diacaaaion of the general ease.
!he statement of the special theorem is rather involved.
this reason, we shall prove it before stating it.
determine aome oondit1ons on y
o•
then obviously there is no solution unleae y
equations 7.1 •
o,
y
fhe pro'blea 1s to
for the ex1 etence of a solution to
In the firet place, if all the L •
31
saying this is that
J'or
mst eatisfy the
(j aa 1. ••• , n).
n
(1, .1 • 1, •••• n) ,
=0 •
Another way ot
linear homogene'Ous
If this condition is tulfillecl,
17
Xi , .... ,
then
~
has detect d • n
are arbitra17.
We
that the system Lx • 7
in this case.
It my happen that non$ of tbe
are not all cero.
proved.
Btq
LJi
bae an inverse, and yet they
This is the case for which the theorem baa not been
In the proof Which followa, this case will be ign.oJ"ed.
We
shall assume at eaoh etep that either all the operators under con­
a1del'&t1on
a~e
sero. or that at 1ea.et one ol them bns e.n illTerae.
If one of the Ljl
The tlrat
~tion
has an invuaa. renu.mbet' so that 1 t is
~l
in e,ystem (5.1) ta
From this we get
n
where
Mu
2). •
Mu T1
is the inverse ot
L.u •
-
i~ Mu_
If this is 8'1.lbat1tut&d into the .other n-1
t.,_i :;_ '
equations, the re8\llt
is
!hia reduce& to
S.t
,.). •
.....51
L
31
.. L K. L..
.fl ~ J.l ~1
(1. J
= 2,
••• • n) ,
•
(J
= 2,
•••, .n) •
Now (5.3) becomes
(5.4)
(J ~ 2. ••• , n) •
sratem (5.4) 1& very aimila.t" to the original qstem (5.1) •
1
tji = 0 , (1, J • 2, ••• , n), the n-1 linear homo•
If all
geneous equations
ma.st be aatiet1ed in order that qatem (5.1) bs.ve a solution.
Ret'•
ea.n be chosen a.rbi trari]¥ and the defect c1 • n - 1 •
1
1
If one of t he L Ji , SAY' L22 , has an inverse, then
x2 , • • • , xn
(J
=J,
••• ,
n) •
Let
Y~
~ ~~
"t~
"•
S,ttem
(5.7) become•
•
T!2. ~
~ 112
J,4fl'
( tJ~
a
3. . . . . . . .a)
.
.
19
(J
=3.
••• • n)
•
Not-e that ~ is a. 11llear homogene.ua function of y~ ud
1
y2
• which in tm'n are lineal' homogeneou.e functions of the original
llow it all
L~1 • 0 , the n-2
e&tisfied in o?d.er that
ue arbi tre.r,- and
d
n
have a solution.
y~ c o must be,
Rere
'=; , ••• ,
x11
= .n-a •
2
I£ one of the
(5.1)
equations
Ilji
bas @on inverse , we contime this process for
stepa or until a. ayatem is obtained where eveq operatoi' is sero.
Iiecall tbat we are asswninc at each step that all operators &l"e eero or
that one of them has an inverse.
After k
steps. we have
(5.9)
and
(5.10)
(J
=k + l,
••• , n) ,
where
(!).U)
Bote that
k
7j
k-1
1s a llJ1e&l' homogeneous function of the 7 J
•
k:l-2
which in turn are linear homogeneous :t\mctions c£ the y J , etc.,
ao that
~ te au.ch a. :t"uncUon
Ae befo:r., if all
~
.
Y'J •
of the original
k
LJi • 0 , we have the
n-k
equations
• 0 (J • lt + 1, ••• , n) whose aat1efa.ot1on is neeessar,r and.
sufficient 1n ord.e~ that system (5.1) ha.ve a solution.
~ ,
1 ••• ,
xJl
are arb1tl"&r7 and the defect
It we an able te continue fo'r'
n•l
d.
=n -
In this case
k •
ateps we have
•
n-l
If Lnn
= o,
y
mu.st sat1sfy the one e<:{l.lation
linear homogeneous equs.tio:n in the y j •
at'"b1trar1ly and we have 4
It
'=D-1
has
=n -
.Also,
n-1
'~n
llh nuv
-= O, a
be chosen
l •
a.n inverse, then
•
(5.13)
• • • • • •
• • • • • •
Jl
~ • ~l 71 .. 1~
Mu. L:!.t ;, •
21
~s
case 1s equivalent to
d.
= 0 since ther-e are no equations
which y
must satisfy and none of the : 1 a.:te a.rbitraq.
We have shown that y mat satisfy a certain system of d.
linear homogeneous. eq'UI'l\.tiona in order that
Lx
=y
have a solution.
Now we want to prove that these equations have the :form
(p • n - 4. + 1. ••• .. n) 1 whe~e the
geneous system zPL
=0
lfeUnit.ign 5.8
11; • Ip w where
If
J
1'
zP
(zP• f)
are solutions of the homo­
•
fhe g§ltf!: QPttator is defined aa follows:
is the identity operator of space
IP
YP •
p, '11e the so~o ope a.tor whteh maps space YJ
ser>o element of
o.
•
1nto the
rp •
Consider the . system zPL • 0 • !his may be Written as
n
~
(5.14)
J*l
z~ LJl
=o
(1 • 1.. • • • • n ) •
a ayatem of equations cons1at1D& eolel.)r of operators,
(5.14) 1s a aet of op~tora ~~
the
n
left members
If eve17 :LJi
e~
=0
•
(J • 1••• • • n)
A solution
ot
Which make each of
to the sero operator.
zP ia arb1traey, Choose
zP • {A~} •
( p • 1. ... • • n) •
Nov
n
t
J-1
z~ 7 J •
!his is the case where d • n
n
t
J-1
A~ 7 J
and the
n
#
yP
(p
= 1,
• • • , a) •
equations which 7
mnat sstiaf,1 are
(5.16)
(zP,
7> • T.p
o (p ~ 1, ••• ,
•
n) ,
a.s vas ahown bef'ore.
Lu_
If
has e.n inverse Mu , we .ha.ve
(5.18)
System (5.18) can be writt·e a
(5.19)
vlth
(1 • 2, ••• , n) ,
1
~Jt
aa before.
If all
l
Ljt• 0 ,
;p
z2 ,
••• ,
...J)
~
a:re arbl tra:ry •
Choose
zP
r;
tzi, ~. ~ , ... , A!}
(p
r.:
2, . . . .. n) •
Then
!his is
~he
caae
a. = n • l , and we
tind that the n...l equations
23
(zP, 'i) • 7P ... Lpl ' \1 r 1 • 0
&8
(p • 2, •• • • n)
a.re the same
(5.5).
.#
If one o"' the
,.l
"'jt
, aq L122 , baa an inverse, then
• •• • n) •
_~f eTeq
Lit =- 0 • then Z~ , • • • ,
Z:
a1"&
arbi tra.ey.
Cho~••
80
tba.t
and
llow
Jtter k
steps , we have
!
oP -- ~
~
., j=t*l
zP4 (-yr-Jrl
-...r-nl)
-~
~
~
(r
= 1• ••• • k ) •
2lf.
and
n
.t
J-r'+l
(5.26)
zPj
.,
1•1 __!...1
Let Hi • ... Lji Mit
Yi
into YJ •
•
* o
(1
=r
+ l, •• • • n) •
fh1e 1s an operator whieh mapa space
With this notation, we can write
zP
r
(5.27)
If all L~t
Lr
.11
Jl
=
.1-1*1
z1:+1 ,
• 0, then
zP xJ
E
••• ,
(5.25) in the
form
(r = 1, ••• , k) •
-'
r
z:
1181' be chosen &:rbit.i"&rU.,-.
Define
zP
(5.28)
=
z~.
tzi,,
•.. • ~·
t.~( J •
( p
We !ll\I.Bt evaluate
zi• ••• • ~~ . uabg (5.27)
~-
(5.29)
n
t
.1*k+l
zPnt•
J
n
(5.}0)
~l = ;k z~ ~l
•
n
t
.1-k+l
k + 1, •••
=k + 1,
,n)} ,
••• • n) •
•
AP~ =
J
-1t
llll
"11:
zt a:,_l + ~1 = ~ ~1 + ~1 •
ln ord.e r to eontime with this process. it is necessaey to use
a much more condensed notation.
Let ue define the symbol
Jf!
\1
the following recursion formula:
, )
( 5.31
.,;ok
~
•
-'Dk
J+2
_n'k
j+a
-'Dk uJ+l
trJ+l ~j + ~+2 HJ
+ ••• + Bj+a HJ + •••
·lf~·~.
25
Tl:m.a
~k is
defined 1n terms .o f
Rj , (a • J+l, ••• , k. :p).
space y J into
yp •
Note that
H~!.
nf
(e =1,
•••• k-j) and
is an operator which maps
~0 give a bette~ idea as to jllst
what the
a!k
ar-e, the first thre& of them are written out ill .full below;
{5.32)
With thia notat ion. w& have
• • • • • •
z~.
•
...-
Jl
~
J-k.-a+l
z~ ttJ, 8.
Iii
~
• • • •••
We shall use indnetion to prove that
(a •
o,
where
1. ••• , k-1).
t
<k
•
!hen
zt- =~8
8
tor
A#swme that it is trne tor s • 0, 1, •••• t-1,
26
zP
lt-t
•
!1
t
Jak-t+l
zP uJ
J ~t
JJPk
vK-t+l
;;pk
~t+2
rP.= oJ
• lc.-t+l ~t + "1t-t+2 ~ + ••• + J ~:t + •••
·rf~t+~t
* Je-t
iJ?k
~1e
•
We ha:.-e proved ths.t
oompl.et•s the indu..otion .
~.
(5.33)
=~~
(a=
o, 1, ••• • ~1) •
Renee,
(5· 3\t)
....
wh ere
(zP -) . t:Pk
• ., • J.
(p
=k+l.
the same as
t he
r;
(5.35)
How
~
••• • n) •
=o,
~'1
-aPk
ut>k
.
+ '""2
7'2 + ••• + ~ ,.k + '1p • 0 •
We want to show t ba.t equ.at t ons (5.34) are
which we had beto~e.
The
r~eion relation for
ta
....r 'I' .....1
Y~8 • 1 ra-1 +. .t1.
8
1
(•
= 1, 2,
••• , k,•
1!
= .,...1,
•• .,a)
•
It we aP.Pl7 the recutta1on relation
• 7pk-3. + uPk
,~3
~
k
a.aa!n. we get
+ ~k . ..kl-.
. 3 + uPk
~3
-"'JI!o-1 •.Jt-l. ~ 7.k.-2 •
J.aeuae lor 1n4uct1on that
Then by
(5.35),
we have
+ uk-e+l .•z-...1 )
'"»-•
+
7&-a
<(.. + ~ ~. + ••• + ~.1 u::; + ·~~· +
+
1
~.. ~:+1> ~=-1
The induction is complete.
~•
'Tp
+ ~k 7k + ~l
Nov for
Y'k:-1 + • • • +
• • k
~J
(5.36) we have
in
~k 11
Ytt-j + • • • +
= O,
(p • k + 1, ••• , n), which la u:actl7 the same system of equations
(zP• '7) • o.
a.s
This ie the cas where d
fba.a, we have proved that
(zP,
~ =
o,
(p
=k + 1,
=n • k •
baa a solution it and only it
:Lx • 7
zP
•••• n). where the
are as defined
above.
Z be rmy solution of
Let
such
z.
!hen certainly
implies that
solution x,
ZL • o •
(zP. 'f) • o, (p • k
+
7> •
1, ••.,
== (Z 1 t'*'i ) • (ZL, i) •
o tor
all
0 for all
n) • which
Converseq, i:f' Lx
L.x • "3' ha.s a aolu·Uon.
(z. y)
Suppose (z,
z
*
7
.baa a.
such that
Bde completes the proof of the fo1loW1ng theorem:
WOBJ!iM 5.1 fhe system
(z. f) •
0
Lx • 7
has a solution if and only 1t
for aJ.l Z SllCh that ZL •
o,
provided that at each step
1n the elimlnation process. either all the operators are zero, or a.t
least one of them has an inverse.
. The . tl:teorem is proba.blr atill tru.e without the restriction on
the existence of inverses.
some examples of the
11
e lhall consider in the next section
in•between • ca e.
We lhall aolve two t:aaplea 1a wbloh x ad y at'• twO""''Ua.-..
~ ,.(o
l-1
(1a) (~)
~
+
00
(6.1)
( oo\o) (~)
~
-1
+
o.)
~
0
(2 1)(xi)
4
QO
(oo oo)
(~)
fhia 1a a quem where· none ot the I.Jl h&ve
ue not aU hro. fheoteua
(6.1) r.acs.a to
~
.
(
•
0)
Q
.
0 0
(ti)
~
• (ri)
r~
an . inver.s~,
iUl4 a\111 t.b.er
5·1 uat not include this • • ·
fU qatea
JO
!his is equ.ivalent to
\
0
0
•
.. xi •
liov
xi
Then either
~
~~
ls u.n1qo.ely determined b7 the fou.:t>th ot these equations .
xi
be detel'm1ned.
or ~ ma;y be chosen arbitrarily and the othe~ will
Also,. we must ha.Te
7~
c
7i
=0
if a. solution is
to exist.
Thus, the
neces~
a.nd autficient concl1t1on for the enstenoe of
a eolution to (6 .1) is
(6.4)
If' this is eattsfied, the solution is
}l
(6..5)
~ •
.. ~
~ •
......1tl'&l"''
~ •
ubUfat7
~
..l
~ • 'i +
2
2
7e • 2xi
J.et ua conat.dw i1Pt the e,retea IL •
•
o • Thia 1a equS;yala.t \o
(6.6)
Ol" t
(6.7)
·z1
1
~1
+ 12 L 2l •
1~2 + 12lt22
*
o
0
Ia •
(6.8)
( ~ ~) · •
z~ z~
SJ'nem 6.7 1• tma
2
( sh zta)(: :)
z!21.
(6 ..9)
zl
22
z11
+
z2
21
(~ ~)(: :) . c2~
~
~
(00)
•0
•10
22
:l (: :).
0
fhl• t'educu to
•O
{6.10)
23:
.·;1
1 )
1
'a
+
0
(
0)
•
0
0 0
zi1•zln•
0
zia • ~ •
o ,
~~
t
• zl-22 • :zfl
•
~~
Z'2: •
••
e.J~b1 i1"8l7
( ~21
z
•
G)
•
o
3)
z1 71 + z2 72
('· 7) •
-(: ~~'h) (~1) ·(~ :) (i)
-z2
21
(r.X~
•
~)
~
·(~)
1t
(a. f)
tha. nidtD.t}¥
~
•
~
•
+
l
7,a
+
ta to equal
we
mll&t
r:)
(~ ~
(~)
•ero tor-
baTe ~ •
0 , then IL • 0
,f
t.ll
,f •
tor all Z
z
•
such that
ZL
o • Convet"-aeJ.7,
nch thU
o•
111
it
ZL • 0 •
!hi• 1a thlla ta• aa.me oo.Ut1on aa (6.4) •
hno•, oample 6.1 aat1•t1e• theor• ;.1.
It &ppeu'a 'YflrT ls.tcel7
that the theorem would bOld tof! all qeteu where the
ma11'1eea and the
~
and 7 J
ate ttnU• d1aena1onal
~
1
are t1n1te
~ecto~a.
Wh•ther
it would be ttue tor othai" ayat•• ia a.n interesti,ng q'l.le&tion,
JiltHfH' 6.2 tb1a .xaaple
ei1Dg\ll&l',
11 oae in vh1oh all the LJi
ret the "f•tem has a. "U1qu.e aol11tJ.on.
~. (:
!)
~ -(~ :)
Lla
.L22
Let
. (~ ~)
. (; .:)
Ah
Z: aDd
1 • • aa 1n e:turple 6.,1 •
!fhe qate Lx • 7
t'edncea to
ext+xi+~+~-
ti
~~+~+~+~• T~
(6.14)
•
1
1
1
2
1
1
l
0
0
0
..-2
0
3 -1
2
..
•
4
•
lroa thla, we knov that qa\ea (6.14) baa a uniqu.e solution for
arb1~l!aq
y •
llhf..a 1& ttla;e e"f'e·n though none ot the
L.1~
baa an
35
It
v.e oomp11to
tot thia oaae,
Z
eertainl¥ ( Z,
i) • o
5.1 appU.ta
1.~
ZL aa
1110.1t
th• preoediDg emm.ple. we find that
••to in ol'der that
be
tot! ar'bl tt"aq y aDd v
to exa.aple
U. • 0
~
fhen
find that theore•
6.2 •
2hia wcgeata the tollovlnc corollat7 to theorem 5.1:
M2HatRX .6 e.J. 1'ho qnem Lx • 7 haa a aola.tloa tffl
7
1t aDd only
~
Su:ppose the. t
1Q11•a that
ZL .tr 9
r
X« •
haa
&tbi tl!!alT
Z ~ 0 •
solu.U.on
to'I' ubitrar,
Z be a:q noto.r BU.ch that U. • 0 • then {z.
y) • o,
1 •
Let
by taeor•
5.1, tor arbitr&l'T 7 • lbt thia 1apl1es I • 0 •
Now np,oae that ZL •
arblttU7 7 and
b • 7
~
z,
o ill;pUea Z • o. !'hen (I• f) • 0 tor
•t1atria6 ZL • o.
Then, b1 theora 5 .1.
baa a aolution for ar-bltr&l'7 7 •
!b.ia cotollary 111 of eour••• tu.OJeot to the aame restriction
a.a theor• 5.1 •
BDlLIOGRAPB.Y
l.
Courant, R. and D. Hilbert. Methoden der Mathema.Uechen
Physik. 2d. ed. Vol. l. Berlin, Julius Springer,
1931. Q69P.
2.
!opicte in Hilbert spa.ee theoey. Notes for graduate seminar.
Corvallis, Oregon State College. l95Q-51. (Mimeographed).