Page 215 - DMTH502_LINEAR_ALGEBRA

Unit 19: Cyclic Subspaces and Annihilators

i–1
polynomial for U (and hence also the characteristic polynomial for U). If we let  = U , i = Notes
i
1,..., k, then the action of U on the ordered basis  = { ,...,  } is
1 k
U   i 1 , i  1,...,k  
1
i
 ... (1)
c

c
U       ... c 
k 0 1 1 2 k 1 k 
...
k
where p = c + c x + + c x + x . The expression for U follows from the fact that p (U) = 0,
k–1
 0 1 k–1 k 
i.e.,
...
U  + c U  + + c U + c  = 0
k–1
k
k–1 1 0
This says that the matrix of U in the ordered basis  is
 0 0 0  0 c  0 
 1 0 0  0 c  
 1 
 0 1 0  0  c 
2
 
      
 0 0 0  1  c 
 k 1
The matrix (2) is called the companion matrix of the monic polynomial p .

Theorem 2: If U is a linear operator on the finite-dimensional space W, then U has a cyclic vector
if and only if there is some ordered basis for W in which U is represented by the companion
matrix of the minimal polynomial for U.
Proof: We have just observed that if U has a cyclic vector, then there is such an ordered basis for
W. Conversely, if we have some ordered basis { ,...,  } for W in which U is represented by the
1 k
companion matrix of its minimal polynomial, it is obvious that  is a cyclic vector for U.
1
Corollary: If A is the companion matrix of a monic polynomial p, then p is both the minimal and
the characteristic polynomial of A.
Proof: One way to see this is to let U be the linear operator on F which is represented by A in the
k
standard ordered basis, and to apply Theorem 1 together with the Cayley-Hamilton theorem.
Another method is to use Theorem 1 to see that p is the minimal polynomial for A and to verify
by a direct calculation that p is the characteristic polynomial for A.
One last comment—if T is any linear operator on the space V and  is any vector in V, then the
operator U which T induces on the cyclic subspace Z(; T) has a cyclic vector, namely, . Thus
Z(; T) has an ordered basis in which U is represented by the companion matrix of p , the

T-annihilator of .
Self Assessment

1. Consider the linear operator T represented by the matrix

 1 1 1 
 
A   1  1  1 

 1 1 0 
 
Show that A is nilpotent. Find the basis vectors that will span the space of the linear
operator T.
2. Let T be a linear operator on the finite dimensional vector space V. Suppose T has a cyclic
vector. Prove that if U is any linear operator which commutes with T, then U is a
polynomial in T.

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