Unformatted text preview: 394 8. MODULES Deﬁnition 8.6.1. The companion matrix of a monic polynomial a.x/ D
x d C ˛d 1 x d 1 C C ˛1 x C ˛0 is the matrix
2
3
00 0
0
˛0
61 0 0
0
˛1 7
6
7
0
˛2 7
60 1 0
6: : :
:7
6: : :: :::
:7
:7
6: :
6
7
::
40 0 0
:0
˛d 2 5
00 0
1
˛d 1
We denote the companion matrix of a.x/ by Ca .
Lemma 8.6.2. Let T be a linear transformation on a ﬁnite dimensional
vector space V over K and let
a.x/ D x d C ˛d 1x d1 C C ˛1 x C ˛0 2 KŒx: The following conditions are equivalent:
(a) V is a cyclic KŒx–module with annihilator ideal generated by
a.x/.
(b) V has a vector v0 such that V D span.fT j v0 W j
0g/ and
a.x/ is the monic polynomial of least degree such that a.T /v0 D
0.
(c) V Š KŒx=.a.x// as KŒx modules.
(d) V has a basis with respect to which the matrix of T is the companion matrix of a.x/. Proof. We already know the equivalence of (a)(c), at least implicitly,
but let us nevertheless prove the equivalence of all four conditions. V
is a cyclic module with generator v0 , if, and only if, V D KŒxv0 D
ff .T /v0 W f .x/ 2 KŒxg D spanfT j v0 W j
0g. Moreover, ann.V / D
ann.v0 / is the principal ideal generated by its monic element of least degree, so ann.V / D .a.x// if, and only if, a.x/ is the polynomial of least
degree such that a.T /v0 D 0. Thus conditions (a) and (b) are equivalent.
If (b) holds, then f .x/ 7! f .x/v0 is a surjective module homomorphism from KŒx to V , and a.x/ is an element of least degree in the kernel
of this map, so generates the kernel. Hence V Š KŒx=.a.x// by the
homomorphism theorem for modules.
In proving that (c) implies (d), we may assume that V is the KŒx–
module KŒx=.a.x//, and that T is the linear transformation
f .x/ C .a.x// 7! xf .x/ C .a.x//: ...
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 Fall '08
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