Lecture 3 JordancanonicalformsCagley HamiltonTheorem

Recall A Kex moduleM a K rectorspace V and a IK linearmap Liv EV

If imgur.co and f is upresentedbyattest A thentheIK linear map 4 K x EndaIV has Ker14 GA Elka

P P A mimicDame fat minimal polynomial for A M is a Kajtorsimmodule

Classification v2 M e Kaya Kya with fr I fr it 17Thrown1

Aon i

ggCei ampanin matrix

for eachg i II a X94 9

This is know as the natural firm41AAN RNF A where I n C ift 7 QeGully with

A 9 CQQ What about alternative ClassificationThenWe factor fax p lx Pssix intodistinct primepowers pix monic inducibleThepi's an the representatives of prime elements in Ikexy Choosethemtobe midEverything is monic so no unit is needed in the factorization

Theocenz V K rector space A GEnd V A 0 ThenV admits a direct sum decomposition V Vpp Up

Furthermore each Up i can be express as a direct sum ofsubmodules isomorphic to

lkcxpyy.ggwith no D us

SIC forme2330

In the special care when IK I charo Eg Kdthen write Pi X d for some L Elk

ofVolEachHyp piece gives a cyclic submodule Wpi.mgimns.mn

m

Id Wpi m has a basis B ou k such that

AWpi B

gg

mxmmatux

d m

PH Wpim is generated by some W e V

Chaim B 3 w A 2 w A a to is abasis

LI x 2mis the minimal polynomial of Wpim

Any dependency will yield a polynomial g with 81A ppgSpay Propositionhim early on binomial Theorem

Alternative 1B din Wpim

Ne A whtcw A 2 CA 2 W yields

A A N w A H w t 4 A a Cw

Also 1 0 w 0 since 9Awp x2

so Alwan has the desired shape D

Corollary Given V A with ga pi pin F B

basis for V with A th g blockdiagonaldamp

2330Furthermore fr pi x ai we have

OAi kid

pygmy

nites em'sI 0

This block decomposition is the Jordancarmical formof thematrix A

sht polynomial

Find V an n din't IK rector space A V V a k linermap We have KEX KCA

X A

Def We define the charade polynomialMA as

Xa dit X In A

Obs If An C meaning C G A G for GEGluckthen Xc XaIndeed Xc out x In G AG

at G x I A G dit G dit xIn A atG

XALemma If 4 Ik Ik is a homomorphismofringsta fields then Xp YUMHere Y extends to Y IK x K x

Prod Exercise

233

CayleyHamilton

Thurs Cayley Hamilton Xp A 0 lie Aa XpWe'll see two proofsVia Rational NormalformsShow X A r o trek sol via A 4 4B bavg.fi e UB

m XA ta Xa

Key Xc q forany g Elka mimic Campanianmatrixforq

Ploof We'll use the Rational Normal form ofA

Mmy I 711921 1er FA

at XE A at i

O

XcaEgg

by lemma belowblocking

se Ia Xa D

Xe f for any manic polynomial felkex

PH By induction n degreeoff

deg f n Xc dit Xiao Xiao Cha adXiao

s dig him in he Xmtam X t tao

x Ogo

ca o

g

mo Im 4 1 Ix

detxInCa is computed by column expansion

Ya x at tix aat 9,9

no9I ii

Extanto

ti

TattiCryaid

X Xegg

t 1 no

I X Ego Go fix 9 tao fix p

IH

A Ishow Xp A v o f v in V

Pick any v e V consider V's Rep u that is therector space spanned by 3h Av Av

We know we can find d withBI v Ar Ad if a23360

basis for V Assume dem Ven

We extend B to a basis B of V Then we get

cats É where a

ISo A Cfpw

Then using the same block decomposition we get

XA XA XAp Fay YA YA 9AluLemma

So Xp A o Xa A Fay A o

Xa A

Fali v50 D

Is The result is true for matrices onany

commutative ring RWe can show this by proving the hymnalXp x X t buyx t t be where bit Z ai

banishes am A ai inside a deafest of Matu IRWe cankick diagonalizable matrices as such set

D

Annalise prod Using Cofactor matrices seeHwa

33ConseguenyoffayleyHamillieCorollary1 Gim AEMatu k 7 CE Matu k with

Ac CA dit IA In

849 XA o dit f A ED at A

CH givesXa A A tan A t tao In to

a In A ItanettaIn C'A

G at'dInAdammuteswithC

So C G C works

Qbs E Cot A cofactor matrix of A with

Col Aij

e ti at Alois

We'llseethis in afuture lecture Awith now is colj rumored

E n 2 A

Xp out EE Ia x a x d be

KEEP tastyXp A A atd A t lad be Iz

L II III 4 1 to aid

C CY A Catd Ia 9 at IdIt ta Coffea o

Nxt Fix R a cumulative ring1330

Corollary Gim AEMatu R 7 CE Matu R with

Ac CA dit IA InPHILCtis the cofactor matrixofA then ACCA dit A InThis fields a polynomial identity on 21Cai So I's validover any ammutative ring D

This corollary gives the general version of CH seeHW Il

Ill H Fray Rang AEMatnaulRwe have Xp A 0

Proof Show Xa A v o f v by usingWactor identity on B X In A BEEB Gt B detBIn Dsetting X A at B XA

Corollary A eMatu B is insertitle ifandonly if detAER

PH G Is char singlet AB detAditB dit In 1

Use AC CA YetA In funCorollary 2

Then A at A C D