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
