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