Math 554 Qualifying Exam.
August, 2009 (JiuKang Yu)
1.
Let
J
e
be the
e
×
e
complex
matrix with
J
j
+1
,j
= 1 for
j
= 1
,...,e

1,
J
i,j
= 0 if
i
6
=
j
+ 1. It
is the socalled
e
×
e
nilpotent Jordan block.
Let
e
1
≥ ··· ≥
e
r
be a decreasing sequence of positive integers and let
A
=
J
e
1
,...,e
r
be the direct
sum of
J
e
1
,...,J
e
r
.
(a)
(8 points) Compute dim ker
A
m
, for
m
≥
0.
(b)
(8 points) Show, without using the structure theorem, that if
J
e
1
,...,e
r
is similar to
J
f
1
,...,f
s
(where
f
1
≥ ··· ≥
f
s
is another decreasing sequence of positive integers), then
r
=
s
and
e
i
=
f
i
for
i
= 1
,...,r
.
(c)
(8 points) What is the Jordan form of
A
2
? It is enough to describe the sizes (and eigenvalues)
of its Jordan blocks.
(d)
(8 points) What is the Jordan form of
A
2
+
A
?
2.
(10 points) Let
F
2
be the ﬁnite ﬁeld
Z
/
2
Z
. How many similarity classes of 3
×
3 invertible
matrices over
F
2
are there? You may use the fact that there are 2,1,2 monic irreducible polynomial
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This document was uploaded on 01/25/2012.
 Spring '09
 Math

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