Math 2270 Assignment 14
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Dylan Zwick
Fall 2012
Section 8.2 1, 2, 3, 4, 5
Section 8.3 1, 2, 3, 9, 10
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-
1
8.2 Graphs and Networks
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Write down the 3 x 3 incidence matrix A for the triangle graph.
The first row has —1 in column 1 and +1 in column 2. What vectors
(xi, 12 13) are in its nullspace? How do you know that (1, 0, 0) is not
in its row space?
The triangle graph looks like:
eclye
I
L
C
—
o
C)
Jc
-I
I/i
0
0
j
I5
•h
4
w
C((
p1feI1o(i/cv
2
7ecq1fe
8.2.2
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-
Write down AT for the triangle graph. Find a vector y in its nullspace.
The components of y are current on the edges how much current is
going around the triangle?
At
(-)
C1fe4
o-1
?h
15
ô
i5
I
J
4-Ae 1-ri,iy1e
3
Q7CC7J
8.2.3
2 from the third equation to find the echelon
Eliminate x and r
matrix U. What tree corresponds to the two nonzero rows of U?
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—Xi
—11
—12
1o
oH
+
+
+
12
fccj
13
=
13
=
b2
1)3
/
1C)LL!
L
C)
-“1
jilt
5
(JOO
3
4
8.2.4
—
) for which Ax = b can be solved, and
3
Choose a vector (b
,b
2
.b
1
another vector b that allows no solution. How are those b’s related
toy= (l,—1,l)?
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>$
0
(1)
o4j
f 0/1
4
5
8.2.5
Choose a vector (fi, f2, f) for which ATy = f can be solved, and
a vector f that allows no solution. How are those f’s related to x =
law.
ct Ire
(1, 1, 1)? The equation ATy = f is Kirchoff’s
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ri1
(
H H
YIYL\
I)
0
4-i)
I)
y(
T
3
(‘)
oik
(YLYJI
o7k
j
C/7
6
f
-
=
Markov Matrices, Population, and Economics
8.3.1
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Find the eigenvalues of this Markov matrix (their sum is the trace):
A—
—
=
I
(.90 .15
.10 .85
Vc
J1
o
-
1
)1c r)qy
75
j
oAer e,yev/U
7
rJ)
0
E
0
0
OL)
-)
f—
c)
bC
0
rCiD
.,
0
fD
,-Th
U
0
c:1
0
rJ
It
1>s
cs-\cr-\
ft
T1
(
ç-J
I.
-
çs
—-
(t)
‘I
S
CN-
>1>
c-H
I
H—
—I-
>
iH
H—
ci
(ti
-,<
r-
<
II
•‘
-I
I
(0
(0
1-
CI)
(0
Cl)
Cl)
(0
oq
(0
j. t:
Q)
8.3.9
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Prove that the square of a Markov matrix is also a Markov matrix.
/3
Ma/koV
a
A
A
A
If
(
A
A
c
5
m
Markov
10
a
a
a
8.3.10 If A
is a Markov matrix, its eigenvalues are 1 and
(
1
The steady state eigenvector is x
11
=