3 LI Uetrixe xponentials

Ai ilo x n

dtAconstantnxn matrix

Df The matrixexponentrletA.is a matrix

defined byocho

identity matrix

etA I t t A i i t A t j t A t ___

志志 t Ah ______ 2

1 this is a generalization of eta0

元吉 f d

us The soI'm to x is given by⼥ t etAE ffay.gl 0 太

To check this y l t eat y o

⻘ let Ai 龀 t t At it A't it ⺮⻔灯

志 it t Ait t ti t fi Pit0 1 Ait t NE t f EAT ___A l i t A it t EAT It ˊ__ ActAi

⼼ 在1 e E I E E

a How to compute etA

z is an efficient way of approximatingversionsetA but not feasible to compute theexact value

0hnnnnnrofetA.coEach column of etA solves 㗠 AtThese columns are a fundamental set ofSolis

such a matrix is called

letn.NL afundane.yyfirst columnof etAaProperties of etA

eltts A et A esA

opsmore generally

et A ēt A I 悲etA__AketA

fetA_ AetA e0A

I

If BC t B B and C do not commute

then generally et Btc x etB etc

Ncturalfnndamentalsetmethod

c.fr a polynomial plz Point RII tR.tt Pmdefine

p A PA t P A it Rn At Pm I

a Recall that p ID P点 P悲⼗ ⼀ tint Pm

plD etA

P 其metA P感 etAt tp_fetAp.nett

PoNetA P.AmetA p_AetA PmetA

lPoA t RA t 1⼈ At Pm I let A

p A etA

p D etA

P1A etA matrix hey identity

TI Cayley_Hamilton Thecharacteristicpolynom.int

of A了个

队 Z det l Z I A the deg n nanSatisfies

只1A o

⼆ Check C H thm for Af

pin det Iz l 9 I 1⼆ 州㠭 到

沿焒州㓵z 3 2 2 2E 6 Z 9 4

22 6 Z 5

叭 A 1弜261 51691

1㗊 11 1㓡1

matrix hey identity t CH than

m PAID etA

0

⼀ Every element of etA solves PAID y 0

1 an in_th order homogeneous linear ODE