Math 2270 Assignment 12
-
Dylan Zwick
Fall 2012
Section 6.5 2, 3, 7, 11, 16
Section 6.6 3, 9, 10, 12, 13
Section 6.7-1,4,6,7,9
-
-
1
6.5
6.5.2
-
Positive Definite Matrices
,A
1
,A
2
,A
3
4 has two positive eigenvalues? Use the test,
Which of A
x < 0, so A
1
1 fails the
don’t compue the )‘s. Find an x so that xTA
test.
-
A3=(’
=(j
2
A
Ai=(?)
)
=(
4
A
1
10
10 \
100)
\
o)
70
5
n
43
I
40
5
5/
/
(
!fl I
7
€4i
Iy/o/- /a/6//
I 7 O
r
e
L
5
7
+
(I)
(L)
/k)•
0
1
+7i)4Y
(f*6L
2
5
i
*
I.
0)
c’.
0)
cj:
U
0)
E
0)
0)
U
-
0)
U
0
r•I
oc)
II
II
.‘-Th
—.
H
c
:
CJD
i:
II
—
0
ci
cN
c-)
c.
0
Th
—
(S
6.5.11
Compute the three upper left determinants of A to establish the
positive definiteness. Verify that their ratios give the second and
third pivots.
-
/2 2 0
A= ( 2 5 3
\\0 3 8
L
-
Z3
5
93
—
O7
Z(I5)3O
L (o-i) -L(1)
A
P
.z
/
J
7
Q3/
9
r{
h ir
7
5
/
5(o4
q/j
6.5.16
A positive definite matrix cannot have a zero (or even worse, a
negative number) on its diagonal. Show that this matrix fails to have
xTAx> 0:
-
(
x x
2
3
X
/4 1 1 /x’
2
x
is not positive when
1 0 2
)(
2
5)
)
f
\X3)
1
(x
,
2
)
3
x
=
(
3
Iy
-
ict
H
-i
19
7)4
1/
I
(o
3
l9y
0
70
1
)(Y
/
3
/(lt)
C(
—
7
6.6 Similar Matrices
-
6.6.3
-
Show that A and B are similar by finding M so that B
A=(
A=(
3)
)
and
B=(
and
B=(1
and
A=(
B=(
(:) ( )7
M’AM:
8)
‘)
).
6 (it)
10)/I
(Id)
M
I
-
i
}
/
(U
) (1 ) 4
4cjJ
/10
(oi
i/(jq
7
Ioj
(d
U
ç
If
iN
c
C
Ii
(N
C
C
1)
0
II
0
--
II
Cr)
C
cjj
C)
0
(D
6.6.10
-
By direct multiplication, find J
2 and J
3 when
Guess the form of
jk•
Set k
=( ).
=
=
0 to find J°. Set k
—ito find J’
A
(
T
k
I / Z/
UA) //
A
(.
ic-j
1
io
I
9
)
(AY
AL1
A)
6.6.12
These Jordan matrices have eigenvalues 0, 0. 0. 0. They have two
eigenvectors (one from each block). But the block sizes don’t match
and they are not similar:
-
/o
1 0 o
0 0 0 0
00011
0 0 0 o)
1 0 o
0 0 1 0
0 0 0 0
0 0 0 0)
o
J—
—
K—
and
—
For any matrix M, compare JM with MK. If they are equal show
that M is not invertible. Then M’JM = K is impossible: J is not
similar to K.
/ Th
I
M
I
I
RH
/
o
LI
1
q
L
L/
I
/
o o
IflLI
,I )Ti
/
k
/
,
1
L3
o
0
0
1
I
0
3I
Q
e
10
(c/C(7
oF M
+Jr
fof
f/M)
6.6.13 Based on Problem 12, what are the five Jordan forms when )
0,0,0,0?
/o
OO
/
ooJ
/
0000
7/
7Ooo
/
QQ
01
/
/
(00
(
oo
0O)
0000
11
=
6.7 Singular Value Decomposition (SVD)
-
6.7.1
,v
1
2 of ATA. Then find
Find the eigenvalues and unit eigenvectors v
/u
Av
:
1 =1
u
-
A=(
(10 20
20 40
and ATA
and AAT
=
(5 15
15 45
1 is a unit eigenvectors of AAT. Complete the matrices
Verify that u
U,,v.
/1
3
SVD
2N
5)
=
1
(u
/u
0
ON
2
v
)T
(0-
5oA
c1d-
0)t1)/0)
-fO
A
-
//o
Lo
12
ij
3
c-
r’J
—‘
11
0
(r\\
11
I.’
-
0
—
6.7.4
Find the eigenvalues and unit eigenvectors of ATA and AAT. Keep
each Av = ou:
-
Fibonacci Matrix
A=(
)
Construct the singular value decomposition and verify that A equals
UZVT.
A
AA AA
1 -3j
HAl
1
-
L
V
13
L
)
-1-
\
11
N
H
I’)
ci
9
cç
><
—
‘H
H
Th
—
Ii
‘I
Compute ATA and AAT and their eigenvalues and unit eigenvec
tors for this A:
6.7.6
-
A
Rectangular Matrix
=
Multiply the matrices UVT to recover A.
(
).
has the same shape as
411 o
I
1-A
(I
o
/
-A)
Th42(/A)
HA
(0, ,3
14
/3/(/-A)5JA
N
11
—
f
—
I I
ci
—
ci
—
—
_-
-cc
\1
—
f’)
I
‘I
ci
—
—
ci
H
r
—.-
——
—
/1
.1
3
I
ôeY
t,
r
0
0’
to
4
a
*i3
4a
Iei
exi3J
oJ
/ 1/r%
6.7.7
-
What is the closest rank-one approximation to that 3 by 2 matrix?
5Q)J
o[l
j
1
L
7-
15
6.7.9
,. , vt-, are orthonormal bases for W.
1
Suppose u
,.
1
u and v
Av =
3 into u
3 to give 1
Construct the matrix A that transforms each v
.
-
.
.
.
-
(1
(I
T
7
U
T
v
C
--
IA
LJ\
/
16