THE UNIVERSITY OF SYDNEY
Math2968 Algebra (Advanced)
Semester 2
Tutorial Solutions Week 13
2008
1.
(for general discussion) Why is a generalised eigenspace invariant with respect to its operator?
Solution
It is sufficient to verify that
W
is
T
invariant where
T
:
V
→
V
is a linear operator and
W
=
ker(
T
−
λ
id)
k
for some positive integer
k
and scalar
λ
. If
v
∈
W
then, since
T
commutes with any
polynomial expression of
T
in the arithmetic of operators (where multiplication is composition),
(
T
−
λ
id)
k
(
T
(
v
)) =
bracketleftbig
(
T
−
λ
id)
k
T
bracketrightbig
(
v
) =
bracketleftbig
T
(
T
−
λ
id)
k
bracketrightbig
(
v
) =
T
bracketleftbig
(
T
−
λ
id)
k
(
v
)
bracketrightbig
=
T
(0) = 0
.
This proves
T
(
W
)
⊆
W
. In particular, any generalised eigenspace of
T
is
T
invariant.
2.
Write down general forms for all 4
×
4 Jordan matrices where the sizes of Jordan blocks are nonde
creasing down the diagonal.
Solution
λ
1
0
0
0
0
λ
2
0
0
0
0
λ
3
0
0
0
0
λ
4
λ
1
0
0
0
0
λ
2
0
0
0
0
λ
3
1
0
0
0
λ
3
λ
1
1
0
0
0
λ
1
0
0
0
0
λ
2
1
0
0
0
λ
2
λ
1
0
0
0
0
λ
2
1
0
0
0
λ
2
1
0
0
0
λ
2
λ
1
0
0
0
λ
1
0
0
0
λ
1
0
0
0
λ
3.
Suppose
T
:
V
→
V
is an operator, dim
V
= 3 and the distinct eigenvalues of
T
are
λ
1
and
λ
2
, both of
whose corresponding eigenspaces are 1dimensional. Describe the possible Jordan forms for
T
.
Solution
λ
1
0
0
0
λ
2
1
0
0
λ
2
λ
2
1
0
0
λ
2
0
0
0
λ
1
λ
2
0
0
0
λ
1
1
0
0
λ
1
λ
1
1
0
0
λ
1
0
0
0
λ
2
4.
Find
e
tA
where
A
is each of the following matrices:
(a)
bracketleftbigg
−
1
0
0
2
bracketrightbigg
(b)
bracketleftbigg
2
−
1
1
2
bracketrightbigg
(c)
∗
bracketleftbigg
5
−
6
3
−
4
bracketrightbigg
(d)
∗
1
0
0
1
1
0
0
1
1
Solution
(a)
bracketleftbigg
e
−
t
0
0
e
2
t
bracketrightbigg
(b)
bracketleftbigg
e
2
t
cos
t
−
e
2
t
sin
t
e
2
t
sin
t
e
2
t
cos
t
bracketrightbigg
(c)
bracketleftbigg
−
e
−
t
+ 2
e
2
t
2
e
−
t
−
2
e
2
t
−
e
−
t
+
e
2
t
2
e
−
t
−
e
2
t
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 One '09
 easdown
 Algebra, Group Theory, Subgroup, Cyclic group, Ker T, internal direct product

Click to edit the document details