LP\  - WEEK f :   El&£A/ VALUES AA/Q EI8EA/ VECTORS*
Uhear operator   ( or  linear enttomorphisn or linear bv&HStormtktoh)   is   4  h'hear nap
op : W —i9 U f vr-fere \4 i'* a vector space  over \K » "fl? or C .
Always,    it koMs     Cf>(l4)^H, cf(ift)~io}.
IA av\o( ib*J are   so  called 6h'ml sub spaces.
The vector subspa.ce, V c~ 14 called //tngfryfr*f£ &u bspace   of  an   operator  cf : 14—? L4 { if-
<fCV) £, 1/
Example. L Cf : TRl*—?TR''     <j>C>) ~ Ay A= 14 ~4   4 -3\     CeusiMer the cctbspace^
1-10    2]     w    r H
-2   3    V   -/ '
We chow th«fc v(s ******** h6r*ce
(D
DeffVufa'oH      The  v*4.t\r(%   o$  ůhe opeméoir Cf : lA—? Id   iVi    -ŕtoe fc>4S/'s   t*    Ci<tl <u^t ...^<f<.) 0^- črhe vetéor šjpace H     i&   a. Jv^tur/'y
cohwhs   are   cooräimkes    of- vectors
^l^Mí-/^*) '*  ÍAe   best* cl.
- ( fat   ^'1 - <Wl žjatople.      Leb  us return*    to the etatofle Y
ft, the  b«s„      « - W^ß)'^|)'^f ■       2^* - Anting
(fie* ) - -3w^f 2*/2r-e3
/%)     « / ^   i   c t I VW,X o 4.
Hence    W Cs also *h cuksp«ce> {or?.
In -this b*s<:*
Ifre have
4-dim   iW*r(**ffc- sub&paoss - <?($ev* vectors.
Let Cf : 14—7 (I b& linear Oft>emkora*0t a e Of <£/*e* one-0lmen&ioM& gj&Lce. Cn2cLi is   inwiriavt bh&H
Cf(ti) - 'XH. 7V*i   for all mtilbpte  of H
~e f£
£/?e  Operator (f>   is totitbfplieAitbq bcf
e*i^&*Vecbo\s    for   q>: Ii-7 14 ff- therx 4   % e \K   such th<zbr
Leb  ftn-b     M - K~ Cf(*)=A* rkete.
A /'s * mvbri*   w l>% h .   ^^ik ^
/'s  eepuiva, \evi-b bo
^ >e) ä^
• 3>
Ts
\ ^
4t/
©
LeitototL     'X e IK   is   4*   eitien value* of ■i he of>e\raío^  Cf> f   f 60 = A> / if 4táôhlfl ff    /S0   u   a toot  &£4&>kBP He ehiHicbehtbi potyvomťač   čfal LA-Ie) .
/f   we know   th-e eigenvalue -theh we ca*  debeirm(v\e    Cot res.founding ^(qevecto^
solvťnq    He s^cbeim   of   ho tot of eneous \ ifiea \r   ecfuaui'eh &
CompHÜLbio*    * f- eigenvalues jot ířene^mC linečtí  ope-ros-kots     cf : IA~> \A .
Ute   Ulc<z   4   beste « of- tí . ^s/i
"R* eľqeHvalueL   of   A e/^H wt/^
<£f- ^.    /CÄe eigenvectors e-f A
(h the   ba^i . of-    \A     bUeh    we    b&ve* \
-
-4
then    , r
/VafcKGŕ?&     &   A A*i<ŕť 5  4re Called s.ľn,)'l<zlr t-f.    bheire. is   <Z HiAČrí'y "P \r<afukir(   auch bfaé
B — T"M P.
/-ŕwwj     šťtHi'faý'   wiuvi'ces   h.4ve the. £4 tne  cha.^icteir\£Írio    po(y vom ťa tf..
=      -p-' C A- %£") P  = ^ P'*• íxíz ÍA~ye\
=  *fe£ ^- A É?) .
®
C&A. wichen e ti 'c    y>o ty W/n< 41  e>f- 4 h   čjeeKí éoir
of.  aHíj   yndbHy.   (jf)^   wh&re ol ťs !pa.si's e>-f- l/i .     (  AH such polynenikč s. 4,1ns.
JExan>p)&- Q_ ftMíi eiqen values. 4** etyeurecéorx of   the operator   (f :   F *—? Ks, (f (1- V» A V /
A - I'4 10 -1   -h O
akt
2-2 2-
1-4-71     O \ - ->5-4?^25Uy
7,   -z z->y
(o, ot p)
S^Sh?   TiM   e^e^l^s ei%e» vector
A- -   í4 'I -1 t