8.6. RATIONAL CANONICAL FORM
403
that
±
A
.x/
is a similarity invariant for
A
; that is, it is unchanged if
A
is re
placed by a similar matrix. Let
V
be an
n
–dimensional vector space over
K
and let
T
2
End
K
.V /
. If
A
is the matrix of
T
with respect to some
basis of
V
, deﬁne
±
T
.x/
D
±
A
.x/
. It follows from the invariance of
±
A
under similarity that
±
T
is well–deﬁned (does not depend on the choice of
basis) and that
±
T
is a similarity invariant for linear transformations. See
Exercise
8.6.5
.
±
T
.x/
is called the
characteristic polynomial of
T
.
Let
A
be the matrix of
T
with respect to some basis of
V
. Consider
the diagonalization of
xE
n
±
A
in Mat
n
.KŒxŁ/
,
P.xE
n
±
A/Q
D
D.x/
D
diag
.1;1;:::;1;a
1
.x/;a
2
.x/;:::;a
s
.x//;
where the
a
i
.x/
are the (monic) invariant factors of
T
. We have
±
T
.x/
D
±
A
.x/
D
det
.xE
n
±
A/
D
det
.P
±
1
/
det
.D.x//
det
.Q
±
1
/:
P
±
1
and
Q
±
1
are invertible matrices in Mat
n
.KŒxŁ/
, so their determinants
are units in
KŒxŁ
, that is nonzero elements of
K
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 Fall '08
 EVERAGE
 Linear Algebra, Algebra, Determinant, Vector Space, Characteristic polynomial, invariant factors, Matn .KŒx/

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