Question 3 A 4 4 matrix

has e Values d 2Az 2

3 2

Ax o

Au v has a solution Truev O is a solution

A is invertible falseT A is not inverp.su

clef A X dads dy oA not invertible one of e values

is O

def A 2 FalsedetA O

algebraic multiplicities are 1 2 Imare d

f As O

It is diagonalitable Nea defectives

depends

A Y rr

T

AI iii iii iiit.IE

Ax o has infinitely many solutionsTrue

Ay o e value

rkA 4 False

ha 4 nullity ofA 3

114 0 algebraic multiplicity 7 geometric multiplicityr l

fr A Xl 1112 111 4 2 2 2 to

2

nullity A I

Atv 4v has at least the linesgindependent blatoons

True

Av Xi 2 V Eespace

a d3 2 Va C espacevi V linearly independent

Question 4 A nxn

ATA invertible A i4vertsble

Pf Def ATA det AT efA

Yet At

Def ATA to E detA o

ODE

Outline

L linear differential operator y

Ly o homogeneous

Solution spare is a verge span

dim order of L h

L first order

Dt A 1 7

Y t a 1 74 0

Integration factorannipply

On bothsides

yl et t a.in y.esyes o solve y

Wronskian of Yi Ye yn

iif my

ji y

If Wto at some point then

y i yn linearly independent

Solve Ly 0 with instantcoefficients

Basic idea is to use

yw erand determine

2nd order 017 E

y t a y ta y o CA

G ar t 112 constants

Guess ylxy e is a solution

Y r e ri

y Mert

r't G r t Gz eo

r ta r ta O

it I THIr

2

Three different cases

Case 1 2 real distinct roots A you or a

ar

Y cxI er y et

Need to check Whonskain

ehxenxwlb.tlHerem

n etr't tr Erith x

k r elk 4 xfo

th FV

All the columns yx7 C y thyL e the't

age 2 One repeated root 9 yar r

af

Find one solution y xy er't r afHow to find the other solution

r f Girt 9 O rt 5 0

D't a Dt AD Dt

DT at Iz y o

DTE 7 2 0

Z X t af 2 x mutiny e4

e x

2 lxg.ee taE.eH.z o

71 7 e o

2 x e C

2 x C ex

Dt't ylx C e Ex

mortgage t

y e t e yx c

ye C

y eatx Ext Ca

Yuk C x E g eEx

y lx e e rn E

Inxs x e erZ

e xerx

Case af Ya two complex roots

r taiZ

j2 l

r 2 t pi K 2 pi2 af p rajaifix e y era

Wronskian ly Yu to

Euler's identity

Y IX d TR 2x EX eP i

e as pxtisinex

Yuki em eex i

ed mspx i sinpx

Yi ed uspx

I i'singbasis ofLy og

Summary If 1 D't 9 Ota

age l G 49270 r't a rt 9 0

hag fur real distinct roots rq

e ek Y is a basis of herl

age 2 Gi Yau o Hattar r 5

em xet

is a basis of kerl

case 3 Uf 492C O r ta rt are

r Ktpir LL pi

e2 nspx e sinpx is a basis

of ke L

Example Y t by't 25 0

It 6h 125 0

3I4iZ

F iffy 8in

ylx1 C e usyx Cee Hsin xx

Higher order equations

The method generalizes toA Y t a y t tant y yan o

I Dnt a D t 1 an D tan

Try solutions fix en

r t a r t ant rt an

Cefn Auxiliary polynomial Pcr

1 r 0 aux equation

per r r r rush ich rankm t met Tonn h ti e he are vomplex

numbersD h D h

o WhaVi the

complex Kleinberg

Case l t real number

D t Y has solutions

et't Xe x'em merix

m linearly independent solupangother real hoots contribute similar solutions

case 2 If r 2tpi it

then I L fi is also a

solution

has solutions

e as px EH sinpxXe Px xensingx

Xm emessex Xm easinpx

z m linearly independent solutions

Collett all the solutions to

D r hior

D himi

p Fhi

we get a basis of Ly o

Ex y lx y o C o

Aux equation r c o

I r't E rtrt Xr E rt i r Eti

hi Ct re Ex r irye i

Lt Pi Complex2 pi conjugate

yix C e thex

13ns x icy sin x

Initial conditions

Ex y t Yy't 4y o yid I Y'co _4

Hux Poly r't Katyrt 212

Yuk C E t c x eH

Y 19 C t Cio IC Iy lot c t z e Y t cafe t 2x

X o

I 2C the XC L Cr f

Y 1 7 e HT G x e

H