Page 223 - DMTH502_LINEAR_ALGEBRA

Unit 20: Cyclic Decomposition and the Rational Form

Proof: We disregard the trivial case V = {0}. To prove (i) and (ii), consider a cyclic decomposition Notes
(13) of V obtained from Theorem 1. As we noted in the proof of the second corollary, p = p. Let
1
U be the restriction of T to Z( ; T). Then U has a cyclic vector and so p is both the minimal
i i i i
polynomial and the characteristic polynomial for U . Therefore, the characteristic polynomial f
i
is the product f = p .... p . That is evident from the block form (1) of unit 17 which the matrix of
1 r
T assumes in a suitable basis. Clearly p = p divides f, and this proves (i). Obviously any prime
1
divisor of p is a prime divisor of f. Conversely, a prime divisor of f = p .... p must divide one of
1 r
the factors p , which in turn divides p .
i 1
Let (14) be the prime factorization of p. We employ the primary decomposition theorem
i r
T
(Theorem 1 of unit 18). It tells us that, if V is the null space of f ( ) , then
1 i
V V 1 ... V k ...(16)
ri
and f is the minimal polynomial of the operator T , obtained by restricting T to the (invariant)
1
i
subspace V . Apply part (ii) of the present theorem to the operator T . Since its minimal polynomial
i i
is a power of the prime f , the characteristic polynomial for T has the form f i d , where d . r
i i i i i
Obviously
dim V
d i i
deg f i
i r
T
and (almost by definition) dim V = nullity f ( ) . Since T is the direct sum of the operators
i i
T , ..., T , the characteristic polynomial f is the product
1 k
f f 1 1 d ...... f k k d .
Corollary: If T is a nilpotent linear operator on a vector space of dimension n, then the
n
characteristic polynomial for T is x .
20.3 The Rational Form

Now let us look at the matrix analogue of the cyclic decomposition theorem. If we have the
operator T and the direct-sum decomposition of Theorem 1, let be the ‘cyclic ordered basis’
i
{ i ,T i ,.....,T i k 1 i }
).
T
Z
for (Z i ; T Hence k denotes the dimension of ( i ; ), that is, the degree of the annihilator
i
p . The matrix of the induced operator T in the ordered basis is the companion matrix of the
i i i
polynomial p . Thus, if we let  be the ordered basis for V which is the union of the arranged
i i
in the order , ......, , then the matrix of T in the ordered basis will be
1 r
A 1 0 ... 0
0 A 2 ... 0
A ...(17)
  
0 0 ... A r
where A is the k k i companion matrix of p . An n × n matrix A, which is the direct sum (17) of
i i i
companion matrices of non-scalar monic polynomials p ,....., p such that p divides p for i = 1,
1 r i + 1 i
..., r 1, will be said to be in rational form. The cyclic decomposition theorem tells us the
following concerning matrices.
Theorem 3: Let F be a field and let B be an n × n matrix over F. The B is similar over the field F to
one and only one matrix which is in rational form.

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