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Matrix 7

The document discusses various mathematical concepts including Cayley's theorem, eigenvalues, and diagonalization of matrices. It emphasizes the relationship between similar matrices and their eigenvalues, as well as the process of diagonalizing a matrix. Additionally, it outlines the calculation of powers of matrices using diagonalization techniques.

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rajagourou
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14 views6 pages

Matrix 7

The document discusses various mathematical concepts including Cayley's theorem, eigenvalues, and diagonalization of matrices. It emphasizes the relationship between similar matrices and their eigenvalues, as well as the process of diagonalizing a matrix. Additionally, it outlines the calculation of powers of matrices using diagonalization techniques.

Uploaded by

rajagourou
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Ch.

a-(7 )2 fod A" inn tenms ot A.


|A- aTl= o

2-32 +36 =o
(a-4) Ca-9) = D.
gn divi ding 2^ dy (2-4) (a-4) gel
(a-4) Ca -4) (a) + a a+4] eA
Quttent remain den
4.

4a+b =qn

giues

b= 4"- a =
4^-9-4)-(
(A- 4) (A- (A) + aA+bT
a A+ b I I by c- I4 -Thorem

=(a4") n+ (14)- )

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w
(S) Venity Cayley .tHailbm theorpm fo
a-(21)3)4

am d find ita imveAAR .


A5
A-4a A + | A * - A as a linean i

(olgmomial 1

2
Henca A Batsfies its own ch-egu, B henee
3
Cayluy tamiltm theorem io vetfed.

A 4A-5I 20 At[A-4 Ta 5

Cosiden (5- 4a -7a+-a -io): (a-a-5)


2-4-52424.
-223
+ &a.
32- |12 -1°
32-122-IS

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a4a5)+a)

+ (A +5){
-(024 +31) (o) + (A+51)
CH sheo.
A +5 I a dinean pstynonial
M

iagemalsatom. )0

De fr Too the
matrices 9 &B
simidan xhue e s t a -ainglan ion
matix Puch tat
12
whic CAse we rite AB. D1

Remark. 2

Any two sii laa matices have ssame eigen )2

valves. 2

Proot. Ret A SB 3
ansingulan
matix PAnch bat
hal
’hene eiot 2

4
B pA P.
Conden B-a pAp- p'aI P|
=rl

|pPl= |2)=1.

A &B hoave same chanaetueshc efuains


&ame
etgun valus.
A &B have

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Pef: iagonalisation of a matix A
mon- singlen
cha proCem of finding p shee
MAM == D
matri Much Hhat
D io a
iagonal matix.
simi dan mabice
Note as A & D ae
tti same eigen
D) A &D have
e A
Values.
Aheorem eigen
squan matix with distinet
sti natix whose cotum
values and is
Mià be
Cam
4 A then A
ane che eigen vectorA trans formatin
diagonalised y tte dimilaniky
See D i dtagonul matix
tt
MA M =p. wshee
oiagonal elements ane thi egen
wshose
valws bt A.
Note.
M io called "modal matrix "
"Spechal mabix 4

P is lalled
ane st dishaeA
No te: T# th chanctuenhe valus
hen aloo chi procs hotdagood provided iti
eigen vectms o4 A a linan hy indepenolent.
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Calculation o4 powa mabiy
we ave D= AM (on diagonalitaha)

A : M DM

A- (MDM) (MD M)
- MD(MM) DM-!
M
= M Dm-!
: M D TD

a' AA =(Mpm)(M DM)


MD*pM
Mp3
= M b2 T D m!

amd

A= MD" M n amg poibue Categu.


Note
a diagonal matix
n
D

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