off htt -...
TRANSCRIPT
Jession15Usesof Jordannormal form v 7,2018
solve y IH AYIH YIH A un matrix
at A4 9 yH II
let f off Htt HittiteHlY HI Xyz t
y IH ee't y CH ctett
HW formula for anyJordanblockHW solve y HI AyHI ingeneralby changingAtoJordannormalform
takingpowers ofmatrix A XIXtJ Jordannormalform
AE XIX Xjr XIX XIX t
I 10 De complexificationmmmm
Decomplexification
L M vectorspaces over dimL c dim Men
multiplicationonly over IR realvector space4,2 decomplexificationof L
f L s M linear f Mp decomplexificationof fouetre outerCl
Thin al l f l i enbasisof L over Q then l en il ie u basisofover IR l inparticulardimply LdimeL
b f L sM linearwithmatrix A Bti C B Crealmatrices inbases 4 em and e i eh thenmatrixof fee lip Min is
Be in bases h em ien lienand h en ie ien
c f L L linear they deff det fl
Proofid any eel e IIauen II buticall u Ii bulu tEadieu
generating
also if liu comb O buticy 0 en em liu indep3baeO Ck
b recalldef ofmatrixelements
Heil fleur1 Cei e'm l B tic
d linearity Iflied flieuf le e n I f Cti B
flat flea f lie if lien1 le i eh ie ie
c Letmatrixof f be B tic
det fr deff fbottomnow I def Btc Efi
B
1 it firstcolumn I deffBetit
c
det Bti c det B i c
deff deffdet fl 2 D
Complexstructury
backfrom4,2 to L Needtoknow J La e t I let ie JEid
De let Vbevectorspace over112Then liuoperatorJ V oVwith J id
is called complex structure on V
Thin let V be a realvectorspacewithcomplexstructure and let
fat ib e a et b j e bedef asmultiplicationwithcomplexnumberstoyielda space F Then T is a complexvector space and I V
Pw clear associativity in follows fwm J id D
note dimzV 2dimeV is evenmatrixofJ canbechosen in somebasis as Fu OE Eu axnunitmatrix
Complexification
Fix realvectorspaceV On HOV take J le ee f ez e
T e ez C ez e I t e i ez so JI id is a complexstructure
Complexification of V is VIV V with this JNote we identified with yo e ko V
i l v O yo 10 4s V s Cei ez le o t O ezI e 07 t i l e z o e tie
i V directsun over112
dimpV dimeVa
f k W def f VE W by f fu re fly fuelmatrices of f and f d the same seigenvaluesthe same
Application
Thm Let f e f lVI with a realvectorspace dimV z l Then fhasan invariant subspaceof dimension1 or 2
Pw If f hasreal eigenvalue s subspacespannedbysomeeigenvectorinvariant
If not alleigenvalues are complexs choose A tin alsoeigenvalueof f d leteigenvectorbe h t i v2 inV
s f v tin flu I ti fly l Htin v ti u l IXY un i luv thwlinearspan of y uz in invariantwider f flu I f ve O
now Tessionlb
V 744k I U canonicallyisomorphic121298
complex vectorspace L s Luz Luz
De The complexconjugatespace Iis def as Luith multiplication El
for CECI EEL
then Luz EL I note ion L andyou14,2can
Why
JE id so has eigenvalues ti ter ett
subspaces L 0 le ec Ef4rdJle ec ilenez1
C Ce ec EC4p19Jleyecl ileukBL s Ce ie
Q z M im