Image Processing Reference
In-Depth Information
3.9
Compute the eigenvalues and eigenvectors of the matrix
2
3
111
021
003
4
5
A
¼
Find, if possible, the matrix M such that M
1
AM is a diagonal matrix.
3.10
Show that the eigenvalues of A
þ a
I are related to the eigenvalues of A by
l(
A
þ a
I
) ¼ l(
A
) þ a
3.11
Find the SVD of the following matrices:
2
4
3
5
,
2
4
3
5
, B
¼
1
10
1
1
1
2
A
¼
3
20
and
C
¼
2
3
34
24
1
35
3.12
Consider the symmetric square matrix
40
0
b
1
b
1
1
A
¼
1
a
1
a
2
(a) Find eigenvalues of A
þ
3I.
(b) For what value of
a
, the matrix A
þ a
I is singular.
3.13
Determine whether the following matrices are (a) positive de
nite, (b) positive
semi-de
nite, (c) negative de
nite, and (d) negative semi-de
nite:
2
4
3
5
, B
¼
2
4
3
5
,
211
140
101
1
1
1
7
6
62
A
¼
and
C
¼
1
4
1
1
1
2
3.14
Consider the following square matrix:
2
1
12
A
¼
(a) Compute A
k
and e
At
using the Cayley
Hamilton theorem.
(b) Find the square roots of the matrix, that is,
-
find all matrices B such that
B
2
¼
A using the Cayley
Hamilton theorem method.
-
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