math 2022 06 week - school of mathematics and statistics ... · (e) put i ike, = m in (kzo) 1st...
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MATH 2022 Week06Worksheet
MATH 2022 Week 6 Worksheet
Qy Putm = [ ?I )
.
Ca) find
me :3 -
- 5.' all - it -
-
ml ! ] --
Cb) By inspection ,from the previous part ,
what are the eigenvalues ? 17-
(c) Find P,P"
and diagonal D
such that M = PDP- l
,
p =
,p
- '=
D=
(d) Find the characteristic polynomial
Xcx ) -- detent - ml =/ IT
=
Ce ) What are the roots of XCX ) ?
X =
Cf I find a general formula for M"
:
M"
= ( PDP - 1)n
= p D~
p- I
=
⑨ Thus find MY =
QY Putm= f I!! ) .
⑨ find
MEI -. fit
mail --
Mf !) -
-
Cb) By inspection ,what are the eigenvalues?
IT-
(c) Find P and diagonal D such
that M = PDP- l
:
P = D=I
(d) Find p- I
:
Ce ) find a general formula for M"
:
M"
=
Cfl Thus find MY =
0-31Put
m = [ II ].
Ca ) Find surd expressions for the
eigenvalues :
Cb) What is the dominant eigenvalueto 3 decimal places ?
x,
= IT-
e) What is the smaller eigenvalueto 3 decimal places ?
t.
-- IT
-
Cd) findM
- I=
(e) Put I .= [ ! ] ,
Ike ,= M In ( kzo )
1st entry of Inrn =
-
1st entry of Ih ,
2nd entry of InSh = -
1st entry of In
Complete the following table to 3 d. p .
To Ii In Is Iy It Is It
lol its
Te I 7-
Sk 3 2-143
Cf) find M [ I , ) =
(g) Read off the dominant eigenvalue I M-
(he) Put To = [ ! ] ,
Int,
= M- '
In = ( III ) Inf
1st entry of Ikk
=-
1st entry of In - I
=
2nd entry of InUk -
1st entry of In
Complete the following table to 3 d. p .
So I , In Is IT
lol Csi:Dth - 2
Uk - o . 75
( i ) find M [
lay) =
( j ) Read off the negative eigenvalue, 17of M
-
Q4/Suppose that M is an invertible
matrix with eigenvector I correspondingto eigenvalue X
.
(a) Prove that X to .
Proof :
Cb ) Prove that I is an eigenvectorfor M
"
corresponding to eigenvalue 5'
.
Proof :