Math 554 Qualifying Examination
August, 1997
In answering any part of a question you may assume the preceding parts.
1.
Let
A
be an 8
×
8 complex matrix satisfying the following conditions (with
I
an 8
×
8 identity matrix):
(i) Rank(
A
+
I
) = 6, rank(
A
+
I
)
2
= 5, and rank(
A
+
I
)
k
= 4 for all
k
≥
3.
(ii) Rank(
A

2
I
) = 7 and rank(
A

2
I
)
k
= 6 for all
k
≥
2.
(iii) Rank(
A

3
I
)
k
= 6 for all
k
≥
1.
Find the Jordan form of
A
.
2.
(a) Prove that any linear transformation
T
of a finitedimensional
C
vector
space
V
can be written in the form
T
=
S
+
N
where
S
is diagonalizable,
N
is
nilpotent, and
SN
=
NS
.
(Hint: Look at the Jordan form.)
(b) Use (without proof) the equivalence of “
S
diagonalizable” and “the minimal
polynomial of
S
has no multiple factors” to show that if
S
is diagonalizable and if
W
is an
S
invariant subspace of
V
then the restriction of
S
to
W
is diagonalizable.
(c) Let
T
=
S
+
N
be as in (a). Let
I
be the identity transformation of
V
. For
any
λ
∈
C
, and any
m >
0, let
W
λ,m
be the kernel of (
T

λI
)
m
. Show that the
restriction of (
S

λI
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 Spring '09
 Math, Linear Algebra, Vector Space, Ring, Identity matrix

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