Plan for today: 5.5 Start 5.3 Learning goals:

Be able to solve a linear 1st order system for which the corresponding matrix has characteristic 1.equation with repeated roots of defect 1 using the eigenvalue method. Be able to identify a phase plane portrait based on information about the eigenvalues of a system 2.

Reminders/Announcements

No OH today - will be on piazza in the evening 1.Read the textbook! 2.

th A S const coef matrix nxn

An distinct real eigenvalues 5 2

A complex conj eigenvalues 5 2

today f repealed eigenu

2 2 matrices

A E IfSeek pair of liu indep sols

Eigen D I repeated malt 2

Eigeu

EA IE E E p2 2 2 1

I III.tl L

Any 2 lin indep vectors are eigenvectorseg Y I'of iv f

x etf'dfifer I x et Lgble

I'd soliu indep

Gen soli x c et f t Get fHad eigenu of multiplicity 2 couldfind 2 lin indep assoc eigenu

Exe x L 41A

deth d E u deaf j go

I 11cg J 116 6

9 lot It 16 0

X lot 125 0

4 5 repeatedWulf 2

Eigenvectors

E 52 E EH

ti Htt 3Gu 4oz 0

samec 4

Take IIOne soin est f

Q How do we find a second line indepsolin

If agenre of malt K has at most

p assoc liu independent eigenvectorsit is called defective Defect k px 2 defect 2 I L

If k liu indep eigenu then CompteEI I complete eigenvalue

Defective eigenvalues of defect LE A X eigenvalue of defect 1

Ned good guess for format of Solin

form y 2g't y o

III fit YeT 1what shouldthese be

It Ate't y t uz 1 E e't It t E e't

Xettty de't yet A v te't 1 yet

t.kz t.eu EA uz e t tu A y te't Ee

O 6

E d EE t I to

If we can find It to satisfyingthen I y t

1 Ez e't will be a Solh

Et t 5 E t.lu E

Method X eigenvalue of defect 1

Solve E dE2 start here

Then Fg I I'Eau

Norte v is an eigenvectore is wet an eigenvector generalized

eigenvector

Bacteraupe

42 x LI 4g 5 eigenvaluedefect LA

E 511 4 44

Looking for 70 so that

f I 45fi 41 Ete

I I II I I INI 43 1883

S gives no restriction

Must satisfy Try a Lthen II Iliff 3 to

Any which is not an eigenvectorwould work Can't take C

flyeigenvector

Ek El f f Lwe found before

Pair of liu indep solsest estffiftffft

x x IGen soli building block

c estf f the ft if