An eigenvalues - Purdue University
Transcript of An eigenvalues - Purdue University
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today f repealed eigenu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