Lectute25ieigeeeveaoroaudeigaevactoreuatri.us)consider the real transformation I ↳ AI from IR

"

to IR?Hails of A as ending vectors around in IR?

A "

acts"ou vectors

→ the action of A is easier ou some"

special vectors!

Deff let A be an eerie matrix .Au eigenvector for A is a hastero vector

EE IR"

anele Heat Aoi = XI for sane scalar XEIR

We call X Hee -eug corresponding to E ; or E the eigenvectorcorresponding to X

.

How can we fail the agavebees? T : 1121>112

"

It> Asi

au eegeeuelue X for A is a number ouch Hai

AI -- XI for sane E. to

run

what does it mean? AI - tsi =D AI - TINI - o

⇒ CA - XIN) I =I for oecue oi -45

So to find thee eegaueleees , we must find all the scalars t

such that the equation(A- > In) se - E

has non -trivial solutions .rewind Heat is the same as faedeeg X's for which A- XIN is not euuertible .

mm

why ? because A-din to invertible then the epuatoou CA - din)E -- ohas a unique solution

, namely I -- o-

the matrix A -XIN is a squared eeeainx . therefore it is not eeeuertoble

if and only if its determinant is zero .

We have proven thee following :

theoremX is an eigenvalue for A if and only if det ( A - TI ) =D

this is called Hee eka'sti

s

① - Show that if 1=0 is an eigenvalue then A is not avertible

<Sol. .> we know that dei CA -XI) -- o is satisfied by the eigenvalues .So in particular it holds for tio

⇒ dei CA - OI) -- dei CA) -- o ⇒ A is not invertible

② if I is an eigenvector of A with eigenvalue X,wheat is ATE ?

< Seb.

AE - XI by detention . Apply A ou both sides .

A- CAI) = AGE) = I A- I = X - XI = HI.

so Asi - HI .

ASE = ACASE) = ACHE)= X'AI = XXI = X'I so A>I = Kai

ASE ACHI)-

- A ( x'E) = KAI --XII = HI .

moral : if I is au eigenvector of A w/ eigenvalue X , then it is also

an eigenvector for AZ with eigenvalue 17 . . eegeuv . for A"

with eigenvalue Snr

③ if d is an eigenvalue for A , teen so is 5h.

< son.

we knew that AE = XI for some Jeff .

we want to show that At = (5410 for some Ito .

How can we find such a E ? auypoess?rn s Te SI . check i

ACSI) - SCAE) = 54×-1=(5-1) ,i .

④ Find the characteristic -epuataeu of A- = & § j'" )

<Sets.we need to fend det ( A - XI) -- O

outta -⇒=/ ÷ / ; exit t- s. 15 E't =

cofactor expansion with respect to eat row

= C- HE - > (3-1)+6] - l - C 12] = - 14-2713-11+64-H - 12 =

= - I (3-1-31+12) - 12 = - X't 412-31 -12=0

If you are asked to find the eigenvalues keen you'd solve their

equation for X.

④ Feud eigenvalues and eigenvectors of A = (I £)ad> eigenvalues . dei CA -SI) -o ⇒ I 'f

>

2?> 1--0rn

⇒ (I- d) (2-x) - 30=0 ⇒ 4=7 and 4=-4

- eigenvectors :

• 1=7 we want a new zero solution to (A - 7-3)I -- o

A- 7I= S ) a- ( I:)

( -6 6 ! o) naff -fig) ⇒ sci - xa -- o

5 - 5 O

E = (I! ) -- x .. ( i ) so any multiple of the vector C! )

is au eigenvector for de 7

• X-

- -4 A - Gate Ata 's - CI f) as ( %)

( Ef ! 8) → ( ft ! f) ⇒ x. tazio E - (Ia) - oaf! )so any multiple of ft ) is an eigenvector for 1=-4 .

A .-→ •

: ::L:: ::÷m:::::÷:"#

A'

-

.

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