Math 2270 - Assignment 11
Dylan Zwick
Fall 2012
Section 6.1 - 2, 3, 5, 16, 17
Section 6.2 - 1, 2, 15, 16, 26
Section 6.4 - 1, 3, 5, 14, 23
1
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6.1.5 Find the eigenvalues of A and B (easy for triangular matrices) and
A+B:
A=(
)
and
B=(
“4
A+B=
1
1
4)•
Eigenvalues of A + B (are equal tot equa
plus eigenvalues of B.
(3A)(i)
)
and
eigenvalues of A
e/j
3,
i-A
0
I
(ix)
3-Al
(\J
I
eIyfVC1
-
53
4
• A. Start with the
2
A
1
6.1.16 The determinant of A equals the product ).
polynomial det(A I) separated into its n factors (always possible).
Then set \ = 0:
—
det(A
—
Al)
sodet(A)=
1
(A
—
l
2
A)(A
—
A). (A
—
A)
--
Check this rule in Example 1 where the Markov matrix has A
and
=
1
.
-J
_oj
-T
JOJ
5
l
ChCJ
4
-b
SL
+
-i-
1
(N
-4-
4
1
4
L
4
c
(TT
+
+
CD
CD
0
CD
CD
t”z
p
II
CD
+
+
II
CJD
II
0
CD
CD
CD
CD
-
CD
CD
CjD
CD
çjq
0
CD
CJDQ
ID
CD
CD
Cç
II’
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I
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H
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>
-
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1—•
N
-J
0
I)
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i
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r
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-----
0—
ii
I
——
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rc)&
cr oc
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s)
rD
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tJ
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N
—.
C
-I.
ci
Tj
A has )
=
1
2 with eigenvector x
=
( ),
( )
and
2
=
2
5 with x
1 to find A. No other matrix has the same ‘s and
use SAS
x’s.
5
=
4
1’
i
d /1-I
(‘I /
oiJ( osJ( o
5/(i(L 3
\05
O 5J(OiJ
(L
8
N
—Q
•
+
N
::S
0
I
1
Ii
ON
I
-
N
n
CJD
Ic
H
I
I
Ix
I
Li
CJD
Ui
6.2.16 (Recommended) Find A and S to diagonalize A in Problem 15.
What is the limit of Ak as k —* oc? What is the limit of SA”S
? In the
1
columns of this limiting matrix you see the
-f A /
7
L
y
-
/o
( oJ
YL-()
1l-L
(9
-
cLL 1
/1vv7
k—
1-3
0
- (
AK
(o_jJ
c
_L
Oo
Lh
ic(I1i(
q
i)
(Cf
t /-1 -l
/
qJLdJ-U11
/9 1
i
ç o
I
1)
LI f
10
-±
I3
(.-, L(
6.2.26 (Recommended) Suppose Ax = x. If \ = 0 then x is in the nulispace.
If -\ 0 then x is in the column space. Those spaces have dimensions
(n r) + r = n. So why doesn’t every square matrix have i-i linearly
independent eigenvectors?
—
c/n
spe a
ifhcc,t
)
3
,f/qce
J
cc/rn4 ce
/ef
/
O/’oyO%q
/
rou
c
7
(Co
(A)(A)
pa(1)
11
6.4 Symmetric Matrices
-
6.4.1 Write A as i’l + N, symmetric matrix plus skew-symmetric matrix:
/1 2 4\
4 3 0
A=
=M+N
6 5)
(
1
For any square matrix, Al
up to A.
(AlT
= A+AT
and N
I,NT=_N).
add
=
1 1
(z 36)
\ o/
:))c
12
(
O3)
Si
¶1
U
‘I)
II
< <
Sj
+
I
II
0
LID
C
CD
CD
CD
CD
LI)
CD
CD
CD
CD
I-.
r
N—
—
Ii.
1
H__)
—
H--
N
C
>
c-rT
-4-
+
-±
-4-
H
S)
-
C
__)
—
I
I
II
I
(0
CJD
N
0
x
0
0
6.4.14 (Recommended) This matrix ii is skew-symmetric and also
0
Then all its eigenvalues are pure imaginary and they also have
for eigenvectors.) Find all
=
for every x so
1. ( Mx =
four eigenvalues from the trace of Al:
=
o
M
1
=
—1
1
—1
iii
O1 1
1 0 —1
—1 1 0
can only have eigenvalues z or —z.
repd
I
I
4
/
-
OkO
15
€iyeflV/U
/
6.4.23 (Recommended) To which of these classes do the matrices A and B
belong: Invertible, orthogonal, projection, permutation, diagonaliz
able, Markov?
(0 0 i\
A=f 010
\\1 00)
/1 1 1
B=( 111
1 1
Which of these factorizations are possible for A and B: LU, QR, SAS’, QAQT?
//
A)
(iJ -J
1
A
q
/izq
/j
A /ko
()J
frje cJ
p
K/ 5J-
2
°u1 I[z(
c oi
C(
Q
t’
J
16
5
5
1