Class notes

WTW256: LU 4.2: EIGENVALUES AND EIGENVECTORS Lecture notes

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Course
WTW 256 - Differential Equations

Institution
University Of Pretoria (UP)

Lecture notes were made while watching the recorded lectures assigned to watch. These notes include theory (theorems) and worked out examples from the lecturer. These specific notes cover Eigenvalues and eigenvectors.

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Uploaded on July 11, 2022
Number of pages 7
Written in 2021/2022
Type Class notes
Professor(s) Ms l mostert
Contains All classes

eigen
eigenvalue
wtw
256
wtw256
conjugate
pairs
matrix

Institution
University of Pretoria (UP)
Course
WTW 256 - Differential Equations

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4. 2. Eigenvalue and Vector
-

method

/
Ai , =
(A) (ki ) (R )
¥!
=

✗ iñi =

Kai ,) =
( r)

Ali , Iik eigenvectors
→
=

,

↳ eigenvalue → nr .
in exponent

Definition

If AÑ :
XÑ ,
then ✗ is called an eigenvalue of the matrix
and Ñ is called
vector
corresponding eigenvector
-

How do we find the
eigens ?
Ali :
✗i
> Because ✗ Isd # put 2 next to

k
,
Ali xñ -
=
o
it + ◦ give it a vector
quality
CA -

•
Ñ :-O -
obvious solution
•
Non-trivial solutions
/ A ✗21=0
-
Determinant

Revision : ↳ polynomial eq ( characteristic 1- A)
•

nun
A >i =
b- will have infinitely many solutions if 1A 1=0

, summary
To find

1) eigenvalues -

solve equation :
det CA -

XT ) :
IA -

✗ 21=0

2) corresponding eigenvector solve system of linear equations
( A- ✗ 2) ñ=j
* Note 1. ✗ =

maybe taro BUTT Ñ≠o !
mm
zero vector is NEVER an eigenvector
2.
Any non zero multiple at an eigenvector is
again
-

an
eigenvector
example
1. ✗
'
=
AF
iii. iii. e- "
:( ; ; / (1) ( ;)
"
A •

A- in
( :} ;) %.IE?naisare
-
-

-
× "
matrix

one , rest zeros

:( 2)
-3

-3 4 : i
/A -
✗
11=(-3-11)/4 -

x) -

l -

6)
( X 3)( x -121 poly equation
'

→
-
-

,

:.\=3# * Don't have
to get in

echelon form !
Xl row
-

l : :/ 111 :( :/
"

of
CA -42 )ñ , •
=
=
. .
.

ii. a-
¥;) :(; :/ - 69426=0 111 Ñ :( 39g ]
-394 b
=
0 (2) a ER

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