Cyclic subspace

Definition

Let T : V → V be a linear transformation of a vector space V and let v be a vector in V . The T -cyclic subspace of V generated by v is the subspace W of V generated by the set of vectors { v , T ( v ) , T 2 ( v ) , … , T r ( v ) , … } . This subspace is denoted by Z ( v ; T ) . If V = Z ( v ; T ) , then v is called a cyclic vector for T .

There is another equivalent definition of cyclic spaces. Let T : V → V be a linear transformation of a finite dimensional vector space over a field F and v be a vector in V . The set of all vectors of the form g ( T ) v , where g ( x ) is a polynomial in the ring F [ x ] of all polynomials in x over F , is the T -cyclic subspace generated by v .

Examples

1. For any vector space V and any linear operator T on V , the T -cyclic subspace generated by the zero vector is the zero-subspace of V .
2. If I is the identity operator then every I -cyclic subspace is one-dimensional.
3. Z ( v ; T ) is one-dimensional if and only if v is a characteristic vector (eigenvector) of T .
4. Let V be the two-dimensional vector space and let T be the linear operator on V represented by the matrix [ 0 1 0 0 ] relative to the standard ordered basis of V . Let v = [ 0 1 ] . Then T v = [ 1 0 ] , T 2 v = 0 , … , T r v = 0 , … . Therefore { v , T ( v ) , T 2 ( v ) , … , T r ( v ) , … } = { [ 0 1 ] , [ 1 0 ] } and so Z ( v ; T ) = V . Thus v is a cyclic vector for T .

Companion matrix

Let T : V → V be a linear transformation of a n -dimensional vector space V over a field F and v be a cyclic vector for T . Then the vectors

form an ordered basis for V . Let the characteristic polynomial for T be

Then

Therefore, relative to the ordered basis B , the operator T is represented by the matrix

This matrix is called the companion matrix of the polynomial p ( x ) .