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Purdue University
MA 353: Linear Algebra II with Applications
Homework 11, due Apr. 9, Some solutions
Sec. 5.4:#16: Let
T
be a linear operator on a ﬁnitedimensional vector
space
V
.
(a) Prove that if the characteristic polynomial of
T
splits, then so
does the characteristic polynomial of the restriction of
T
to any
T
invariant subspace of
V
.
Proof.
Let
f
(
t
) be the characteristic polynomial of
T
. Then since
it splits, it is written as
f
(
t
) = (

1)
n
(
t

λ
1
)(
t

λ
2
)
···
(
t

λ
n
)
for some
λ
i
, where
n
= dim
V
. (Here
λ
i
are not necessarily dis
tinct.) If
T
W
is the restriction of
T
to a
T
invariant subspace
W
,
and
g
(
t
) is the characteristic polynomial of
T
W
, then by Theorem
5.22, we know that
g
(
t
) divides
f
(
t
). Namely,
g
(
t
) is a factor of
f
(
t
),
i.e.
g
(
t
) = (

1)
k
(
t

μ
1
)(
t

μ
2
)
···
(
t

μ
k
)
where
k
= dim
W
and each
μ
i
is
λ
j
for some
j
. Hence in particular
g
(
t
) splits.
±
(b) Deduce that if the characteristic polynomial of
T
splits, then any
nontrivial
T
invariant subspace of
V
contains an eigenvector of
T
.
Proof.
Let
W
be a nontrivial
T
invariant subspace of
V
. By the
previous part, the characteristic polynomial
g
(
t
) is written as
g
(
t
) = (

1)
k
(
t

μ
1
)(
t

μ
2
)
···
(
t

μ
k
)
where
k
= dim
W
and for each
i
we have
μ
i
=
λ
j
for some
j
. So
for some
j
,
λ
j
is an eigenvalue of
T
W
. Namely for some nonzero
w
∈
W
, we have
T
W
(
w
) =
λ
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 Spring '08
 Staff
 Linear Algebra, Algebra, Vector Space

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