7 4 Continued

Recall i PNZ de t Z I A characteristic poly

Every element of etAsatisfies an n th order ODE

PAID y 0

Recall the natural fundamental set of sins to PAID yo

No It M l t Nun t is defined as i

Njltl Solves PAID Nj 0 Nj ojhioktjko.l.irn 1

Then the Sok to PAID y 0 y lo yo y花 ⽕is y it y NoIt ty ⼼⼼ t tyu Nn.lt

Initial conditions for etA

etAL e ⼆ I

getA𠃊 ⼆ At It A____

灬

志州 ⼼ etAl⼆ ⼼

etA No It It N.lt 1 A t tNn.lt A

⼆ Compute etA for A II last time PA12 22 6Z 5 Z 1 E 5

Find natural funda set No It N.lt

PAID 0 y 6 y t 5 0

roots of pn.IE i n 1 n 5both simple

General sol n to PAID yo y HI ⼆ Get the5 t

y It Get 5 GetFor No l t No10 1 以 10 0

I C ⼗ G

0 G 5 G

G É

No It f e t_test2 ⼆ ⼀ 本

For N.lt N 10 0 N 10 I

O C ⼗ G __⼆ C 5 G

G 41 z

1

N.lt f e t f e5t

4

e tA No It It N.lt A

lie t.ie 541691

tft.ettt.it 㕣

fettt fettjt

tettteTtettte5tfet te5t.fet.itiet tte5t tetties ttetttettettte.tl

了 5Eigenmethod

DIAneigen ofa.ru xn matrix A is

a complex number ⼊ such that thereexists

a non Zeo vector Ǔ w

Air ⼊ i

Such a vector Ū is called an eigeyassociated w ⼊ I⼊ Ǔ is an eige

ofA.coA i ⼊Ǔ ⼊ I A i 0 det MI A 0The eigenvalues of A are the roots of PA It

⼆ Find eigenvalues and eigenvectors of A ⼆经引

I hit time plz E 6Z 5 Z 1 2 5

eigenvalues ⼊ 1 ⼊z 5

For ⼋ 1 A Ū⼆ ⼊Ū Ǔ 1号

1㓵忧 州了 n t 2 n v

2 u 2 uChoose u 1

u n.tn then Vz 12 u 3 n n l Ni u 2 hlSame thing

For ⼼ 5经 引仙 5 N

3 V 2⼉ 5ˇ

heyerr

Choose y y

2V z 2 uthen n 1

1⻔any nonzero multipleof an eigenvector is also an eigenvector

If ⼊ Ǔ is an eisenpair of A then

Ǒ It e t iis a so I n to

惑 ⼆ AT

To check this

表 ètū n e N it

A let Ǔ et A i e ⼊Ǔ

Diagonalizablematrix_Iflthaseigenpairsln.liˊˊˊ lin Ǚ such

that i I are linearly independent then

A V D Vwhere

V Ii ii in 叫 ⼋⻔Such a matrix A is calledadiagon

matrix.iocheck A EV Dr or AV VDN Ali I i A i.AE

⼀ Ai

⼊⼼ ⼋Ǔz ⼊⼼

VD li.in in ⼋⼋ ⻔1 in innit

ifit Ǚ ⼊⻔

on If A V D V then etA Vet Dr

N VD_tEVDkvktiw.esDiagonalite API and compute et

A

Eisenpairs ill.IN 15 111A V D V v 1 I D 16⽉ě IN 佔断

illicitly

Tae19年11 i Hit if

it _ětět

i _test et test

州 111 1 ⻔A 111 1 I ⼀