Assignment 12 Math 2270
Transcript of Assignment 12 Math 2270
Math 2270 - Assignment 12
Dylan Zwick
Fall 2012
Section 6.5 - 2, 3, 7, 11, 16Section 6.6 - 3, 9, 10, 12, 13
Section 6.7-1,4,6,7,9
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6.5 - Positive Definite Matrices
6.5.2 - Which of A1,A2,A3,A4 has two positive eigenvalues? Use the test,don’t compue the )‘s. Find an x so that xTA1x < 0, so A1 fails thetest.
10 \Ai=(?) A2=(j
)A3=(’
100)10 \
A4=(1 o)
70
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Th
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c-)0
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c.
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CJD
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6.5.11 - Compute the three upper left determinants of A to establish thepositive definiteness. Verify that their ratios give the second andthird pivots.
/2 2 0A= ( 2 5 3
\\0 3 8
L-
5 Z393 —
O7
L (o-i) -L(1) Z(I5)3O
A P
.z / 7J
Q3/
9
r{ 7 / 5(o4 q/j
h ir 5
6.5.16 - A positive definite matrix cannot have a zero (or even worse, anegative number) on its diagonal. Show that this matrix fails to havexTAx> 0:
/4 1 1 /x’
( x x2 X3 ) ( 1 0 2 f x2 ) is not positive when2 5) \X3)
(x1,x2,x3)= ( —
C( 7Iy3
- l9y3
ict 19
H -i 7)4 1/ / 3
I 0 / (lt)
(o 170)(Y
6.6 - Similar Matrices
6.6.3 - Show that A and B are similar by finding M so that B M’AM:
A=( 3) and B=( 8)A=( ) and B=(1 ‘)A=( and B=( ).
10)/I(:) ( )7 6 (it)
M(Id)
I /10- i } ) (1 ) 4/ 4cjJ
(U(oii/(jq (d
7
Ioj
iN
1)C)
Ccj
j
C(D
UC
CCr
)0
IIIi
-- 0
If
c(N
II
ç0
6.6.10 - By direct multiplication, find J2 and J3 when
=( ).Guess the form of jk• Set k = 0 to find J°. Set k = —ito find J’
A
A I / Z/ (AY AL1( UA) // A)
T )k (. 1 ic-j
io
I
9
6.6.12 - These Jordan matrices have eigenvalues 0, 0. 0. 0. They have twoeigenvectors (one from each block). But the block sizes don’t matchand they are not similar:
o 1 0 oJ—
0 0 0 0— 00011
0 0 0 o)
/o 1 0 oK—
0 0 1 0
0 0 0 0)— 0 0 0 0
For any matrix M, compare JM with MK. If they are equal showthat M is not invertible. Then M’JM = K is impossible: J is notsimilar to K.
and
I / ThL 3 1 L/
M , / o
I I LI 1 L q IRH IflLI
o o /,I )Ti
/k / o
0 10 I 3I 0Q
e (c/C(7
10 oF M+Jr fof f/M)
6.6.13 Based on Problem 12, what are the five Jordan forms when ) =
0,0,0,0?
/ OO
/ o ooJ / 0000
7/
7Ooo/ QQ
01
/ /(00
( oo0O)
0000
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6.7 - Singular Value Decomposition (SVD)
6.7.1 - Find the eigenvalues and unit eigenvectors v1,v2 of ATA. Then findu1 = Av1/u1:
Verify that u1U,,v.
is a unit eigenvectors of AAT. Complete the matrices
SVD
5oA
//o
Lo
A=( and ATA(10 20
20 40and AAT =
(515
1545
/1 2N3 5) = ( u1
c1d-
(0-
/u ON0
v2)T
ij0)t1)/0)-fO -
A
12
c-
3
r’J
—‘
11I.’
(r\\
011
-
0—
6.7.4 - Find the eigenvalues and unit eigenvectors of ATA and AAT. Keepeach Av = ou:
Fibonacci Matrix A=( )
1 -3jHAl
A
Construct the singular value decomposition and verify that A equalsUZVT.
AA AA
1 -
L
L
)V
13
cç
ci I’)
11
9
H N
><
-1-
\
Th‘I
—
—
H
Ii
‘H
6.7.6 - Compute ATA and AAT and their eigenvalues and unit eigenvectors for this A:
Rectangular Matrix A= ( ).
Multiply the matrices UVT to recover A. has the same shape as
411 o
I 1-A
(I -A) Th42(/A)o / HA
(- /3/(/-A)5JA
0, ,3
14
ci
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11
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—
/1 .1
4
t, a Iei oJr *i3 / 1/r%
I3 ôeY 4a exi3J
0 0’
to
6.7.7 - What is the closest rank-one approximation to that 3 by 2 matrix?
5Q)J
o[l
1
j
L
7-
15
6.7.9 - Suppose u1,. . . u and v1,. . . , vt-, are orthonormal bases for W.Construct the matrix A that transforms each v3 into u3 to give Av1 =
(1
-
(I
7
vT
/
T
U
C-- IA
LJ\
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