Page 221 - DMTH502_LINEAR_ALGEBRA

Unit 20: Cyclic Decomposition and the Rational Form

Notes
Apply Step 2 with 1 ,...., k replaced by 1 ,....., k and with k .
Conclusion: p divides each p with i < k.
k i
Step 4: The number r and the polynomials p , ... , p are uniquely determined by the conditions
1 r
of Theorem 1.
Suppose that in addition to the vectors 1 ,...., r in Theorem 1 we have non-zero vectors
,...., r with respective T-annihilators g ,....,g such that
1 1 r
T
V W 0 Z ( ; ) ... Z ( ; ) ...(11)
T
i
i
g divides g , k = 2, ..... , s.
k k – 1
We shall show that r = s and p = g for each i.
i i
It is very easy to see that p = g . The polynomial g is determined from (11) as the T-conductor of
1 1 1
V into W . Let S(V; W ) be the collection of polynomials f such that f is in W for each in V, i.e.,
0 0 0
polynomials f such that the range of f(T) is contained in W . Then S(V; W ) is a non-zero ideal in
0 0
the polynomial algebra. The polynomial g is the monic generator of that ideal, for this reason.
1
Each in V has the form
f ... f
0 1 1 s s
and so
s
g g g f .
1 1 0 1 i i
1
Since each g divides g , we have g = 0 for all i and g = g is in W . Thus g is in S(V; W ). Since
i 1 1 i 1 1 0 0 1 0
g is the monic polynomial of least degree which sends into W we see that g is the monic
1 1 0 1
polynomial of least degree in the ideal S(V; W ). By the same argument, p is the generator of
0 1
that ideal, so p = g .
1 1
If f is a polynomial and W is a subspace of V, we shall employ the shorthand fW for the set of all
vectors f with in W. We have left to the exercises the proofs of the following three facts.
T
(
T
1. f Z ( ; ) Z f ; ).
2. If V V 1 ... V where each V is invariant under T, then fV fV 1 ... fV k .
,
k
i
3. If and have the same T-annihilator, then f and f have the same T-annihilator and
(therefore)
Z
T
T
dim ( f ; ) dim ( f ; ).
Z
Now, we proceed by induction to show that r = s and p = g for i = 2, ...., r. The argument consists
i i
of counting dimensions in the right way. We shall give the proof that if r 2 the p = g , and from
2 2
that the induction should be clear. Suppose that r 2. Then
V
Z
dim W 0 dim ( 1 ; T ) dim .
Since we know that p = g , we know that Z( ; T) and Z( ; T) have the same dimension.
1 1 1 1
Therefore,
Z
dim W dim ( ; T ) dim .
V
0 1

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