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Unformatted text preview: 409 8.7. JORDAN CANONICAL FORM The companion matrix of p.x/ is the 1–by–1 matrix Œ , and N is the
1–by–1 matrix Œ1. The matrix of T1 with respect to B is
2
3
00
00
61
0
0 07
6
7
0 07
60 1
6: : :
7
Jm . / D 6 : :
:: ::: : : 7 :
: :7
::
6: :
6
7
40 0 0 : : :
05
000
1 Deﬁnition 8.7.1. The matrix Jm . / is called the Jordan block of size m
with eigenvalue . Lemma 8.7.2. Let T be a linear transformation of a vector space V over
a ﬁeld K . V has an ordered basis with respect to which the matrix of T
is the Jordan block Jm . / if, and only if, V is a cyclic KŒx–module with
period .x
/m . Proof. We have seen that if V is a cyclic KŒx–module with generator v0
and period .x
/m , then
B D .v0 ; .T /v0 ; : : : ; .T /m 1 v0 / is an ordered basis of V such that ŒT B D Jm . /.
Conversely, suppose that B D .v0 ; v1 ; : : : ; vm / is an ordered basis of
V such that ŒT B D Jm . /. The matrix of T
with respect to B is
the Jordan block Jm .0/ with zeros on the main diagonal. It follows that
.T
/k v0 D vk for 0 Ä k Ä m 1, while .T
/m v0 D 0. Therefore,
V is cyclic with generator v0 and period .x
/m .
I
Suppose that the characteristic polynomial of T factors into linear
factors in KŒx. Then the elementary divisors of T are all of the form
.x
/m . Therefore, in the elementary divisor decomposition
.T; V / D .T1 ; V1 / ˚ ˚ .T t ; V t /; each summand Vi has a basis with respect to which the matrix of Ti is a
Jordan block. Hence V has a basis with respect to which the matrix of T
is block diagonal with Jordan blocks on the diagonal. ...
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 Fall '08
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 Algebra

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