Mnitairq  wet   selfad^oťnb   operators   #h a space, U have 41 Mys ůůtW oj basis  iv\ H fom&et b^j   ecqeh vectors.    generally -this ťs heb true for  other operators,
an \     / v
This operator   fitfs ttobeŕfehraftie.   (L of atyebniľc.
toulttpliatcß 2 4lMt qeovn&tiric* tou liľplľaéc^ 4 abau's ih   R* famed        eíqenvetbo**. Thtirf*
cf cannot  have   ť» #h„ hasť^u
HotívAhiok\     Tor^pevatortcf With the property -čh&ů   the 9(4m of atyebwa nulbťph'tCtfeš <ž>f l-U   ei'fe*ua/ífe&   ťs &fwt   re> the eťihi&picťoto of spzce.   oh whťch cf> is 4efihee(    we wnb
bo fth* a tes/s (n urhfeh the vnztrtY of <p /'s   #s ôfm /oh 4*   possible.
This simple fotm is JORPAA/ CAbot/iCAl Fo^ (We will mibe 7ÔF.)
ľ>ef-ľmtľoh of JCF [■f)    Tlotáah cg II   X, 00 is -the mitrti
o o> O
0
/4 toa-ŕť/'x  Ts   in  lortfAh canonical -f-otvn (f ít cs      block,  (diagonál weh  Oorohh ccIk
£ ta tuple.
®
ôhaľn   for  the e.ľq^nmlne %  č>f <&s&ah cpefä
6or   Cf> : IA —> IA    /'s    ^ ce><ft4<?hc<s~ u1t(4H^                höh &ero   vectors, f trom l/[
(((-        Hi ~ 0
b
l^e im'// «m'fce t'fc £otoetľm<2£ ih ehe fol/ov^u^
ft
Cf-W                              Wo*,     <ŕ-lM (f-Ktf lá*.--> Wfc , —> • ■  —^ -? u* -> «1 —?0
The»     Y - C «fí 0Z/     % J o, 14 c$> #h Mvturitwt basts   éhe   makmi*   <of   f JV : V—? V /i
/y 4 o
6> o >
TWf     (w^-f«/   ^or -the au*putehbrts) The conditio*e  -finovi  £Jie rtef-ťhtbn   of Aehetfy
Hence,    <f(u<) e í/*, coktzefuen-tt^
pehdevit ( 11 <&h be chown i'núitíchľon Mth respe^é- bo t t iwt- I mII Me-é 0*0 ik") £0   <x s 6*/,.- ís   <2 iaarri k
	1	0 .	0
í Ö	>		
0			0 ■
	1	■ ■	
\<?		0	
d orda n Theorem
M be- * sector jf^ece *f fitťmen&ton h apd  (f \ (Aa* operator, suppose
éhair th^e suht of alqebmcc multiplicities of  all eigenvalues of <^> is ofu<zl to h.
Theh    there Cs   « Itazi^ o<    in   14 euch ťha ér
This   HtA-ítríy is d&bermfto&rf uHiýéte/cj
Uf>   to   6he   Oiroter &f    Votáah Col(x .
TfeniarL The basis o( from theorem <zhe</& Íq    fowled        Chai'hQ.   ú&  e-iqentotliiesL <=>f~
^}oWl4.h   Theorem   e>ve\ŕ  Cotop/ex hi4i*tbors If   Cf : IA-7 U ti ľs 4 čoňip/ey. i/ectoir
gpzc& t   we CA*  omi'-t   the assu rnjptioftM
oh the toulítplľcťŕte* , *i*ce> lí í2c al^x
of äeq^ee h lits * iroots incl^i^ nv/éi-folľcítf'es.)
(E)
Haitis   version of   ÜoirafaH The<ote?m ^ Le-b A bc^Qhfc *h)~ v*atrt> IK =-~R0\r£,
Gumose   thah  the H i*«fé9pti#'érľ*£
of (ts   eigenvalues   /i  &fu^l áo h. ZT/fet?
A ľs   eľtoľte^ ío ahviuŕť* XI ľh /: ^.
there ľs   4 refuted huzéin'* V ££*<zh ůkzé-
-3  = T"1 A T> The nuikVii J (s dt&éeirmľyiGBŕ unľfHelq up 60  éhe  order   of   3owfah ce/U.
~R&V)rú.\rí. ®The HVLbrť* T ľs hoir 4&uermľn&>/ ahifuelq I
(S) Ove\r d! We özn omi'b ůh& tissuinptľoh. Troof   :    7<ŕ>rafoJt Theory*   =-> Ma&ľy teuton
in  Ot^TgôrC.    Then   f& äef( pes   <z h
which satisfies* the c?s£c< nation of lorfa^ Theorem.    Then  ťheire is> ^ b&eis * in
/ tí Mas
I.
3 -<riW*)K/r><
flo&te,   €, = (^1 is the sbzučet basis,
FIMĎIMG      1CF     FOR   OPERA TPR$> AM£)
HATRICBS   ///    DlMEÁ/SlQH    3  AA/D fy
Tule 1
there
\ one i c
eiqehvalues>   af eur Cfembor ce>   (h>a.érty A)
i
e-vetu-   as   tnafiq times as   íás. pm a/<febmť0
A •» /"5   *•    3 \    (á&ř-^érer/itrc polywtriíal í's
6  fs- 10) E=i'qen values.
a/^. tnuli.   2( qeon. mult, 2 e i^«en Kec éor&     ^ ^(zt t(-g) r
®
h ůhe   tas/s    ^ Ä (\ŕu i/^ v±) fm
This   is  a Hdhn\   in   dCF   With 3 of 11
>, - 2     alf* null,  iŕ  qeow- null. 1
eiqefi vector [1( o£o)T
/4^h^/^   t#    ^ule1   the oiC^om.I of ~JCF
1        bh\ree cells t oůh*rWse. é he qeom. mult* of ^1 would   be* 2-. So   3tF Jms  to h&
/2_	
o	
	oh /
e
(D
dve   IooIlChq   for & boat's.   « f h vvhcc\r\
et
	0 t?
	
	0 1
J
T-o\t  -b^^Cíqey\\ŕčili4<s *Z   \&& hav<e. e>iqe^vecbor
Vi   in oc .     Tor   the <siyc?ni/^/^e  -Y iťô
u> f-ŕhat  čĺ cteín  e>£  the length 2.-far *ffii*te>í
T4& firs6   vector       this chain is &h eťqeft ve~cbo\r   \ri •    The  ^e**** vector is    |/S   $uoh éhaé
2		c
		1 >
/y^rď wer
__. ©
Tittle. 2. f
The  tnuinlpeir of-   Oerfáh c&Ds.   fot the.
e-iq e rvalue  > ,'s   eefital  to ehe qeoinetn'c
foul-tiplia'irc/ of %
EW\aWLbCoiň : Chain
f- Krt <?-W f-M
V«. —> Vfc.f - -. . —> f2 —> k, —? o
Byam jole. 1~     y —? R5 A x
Qcc&teti^   /é**   IZtdb Z j OCF >&>o #kUj tme.
That frkej we have ío looí f-ov a basis <* fa the of £ čhzih p-f   length 3 ${#irhi*q
Mt h   the vector   ^ t
I
We   so/t/e &fu4Ů/OHs>
(A - Z£) (42  - ^
(A - ZET) *3    - *2.
<i?//e possible   ^plutiohs /'s
1
Ih the basis oi ~ (tii, KZf Hz )
Im, ( A- 2a)     r*-, upA^e.
rfccoKttnq   ée   TV/es Y atrsť 2.   <Z 3CF f or f Cwbzľhs     ÍWO    ce//s ( Ohe   0-f   Ott^
Ute to find a. basis «   which eonsicés.
č>£    hwo   ehaľhs    f oř the eigenvalues 2.^ ohz*   o£   lehfúh   2   4 há <oh& of l&ntjth f^Jp (oh/^ <síqeh vector} /   Urhí&h Iľtxstus/c^
H/e hat/e to  fiŕhd aii e.(qetn\/<2ctor Uťhľôh ts ah the b&jľnnťnff  &f £h<s> &htun of-
We loot -for Ct   in é h e £om
au -t by
4Q 4 vector fay* which   the <z?h<tt(om hä£   Q,   šolHtt&h .
1 H O -/ -2 O -2 -<4 O
As
«. (2      -Z^tfyid   compute 4$ & šoluúľoii
0hc  y osu bi lit <q    í $> s (-4, 1t ij^
'—-—'' <s nofc 4 multiple a einfror ow length 2 r *■<
i. e v -
(v)
2-	1	0
0		0
0	0	Z
"Zt'/Kď a bast's <*
the opemkoir  (f> :
6
•ŕ?
-2
J" £
1
in which
7-8
3/
tíľnér : £ľfl&nvetnej Cite. 2 a f afy-tokli 3 tfl^   A ,