MAS 3114—Practice Problem Set #3
1.
(a) Define what it means for two
n
×
n
matrices
A
and
B
to be similar.
(b) Let
A
=
bracketleftbigg
1

4
2

3
bracketrightbigg
. Find a diagonal matrix
D
(with complex entries) such
that
A
is similar to
D
.
(c) Let
A
=
bracketleftbigg
1

4
2

3
bracketrightbigg
. Find
a, b
∈
R
such that
A
is similar to
C
=
bracketleftbigg
a

b
b
a
bracketrightbigg
.
2. Let
A
=
bracketleftbigg
1
2
3
4
bracketrightbigg
.
(a) Carry out two iterations of the power method for calculating the dominant
eigenvalue of
A
. Use
vectore
1
as your starting vector.
(b) Carry out two iterations of the inverse power method for calculating the
eigenvalue of
A
which is closest to 0. Use
vectore
1
as your starting vector.
3. Define a sequence of vectors
vectorv
k
=
bracketleftbigg
x
k
y
k
bracketrightbigg
by setting
vectorv
0
=
bracketleftbigg
1
0
bracketrightbigg
and
vectorv
k
+1
=
Avectorv
k
for
k
≥
0, where
A
=
bracketleftbigg
1
.
7

.
3

1
.
2
.
8
bracketrightbigg
.
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 Fall '08
 Olson
 Linear Algebra, Matrices, Diagonal matrix, Eigenvalue algorithm

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