Cyclic subspace


In mathematics, in linear algebra and functional analysis, a cyclic subspace is a certain special subspace of a vector space associated with a vector in the vector space and a linear transformation of the vector space. The cyclic subspace associated with a vector v in a vector space V and a linear transformation T of V is called the T-cyclic subspace generated by v. The concept of a cyclic subspace is a basic component in the formulation of the cyclic decomposition theorem in linear algebra.

Definition

Let be a linear transformation of a vector space and let be a vector in. The -cyclic subspace of generated by is the subspace of generated by the set of vectors. This subspace is denoted by. In the case when is a topological vector space, is called a cyclic vector for if is dense in. For the particular case of finite dimensional spaces, this is equivalent to saying that is the whole space.

There is another equivalent definition of cyclic spaces. Let be a linear transformation of a topological vector space over a field and be a vector in. The set of all vectors of the form, where is a polynomial in the ring of all polynomials in over, is the -cyclic subspace generated by.
The subspace is an invariant subspace for, in the sense that.

Examples

  1. For any vector space and any linear operator on, the -cyclic subspace generated by the zero vector is the zero-subspace of.
  2. If is the identity operator then every -cyclic subspace is one-dimensional.
  3. is one-dimensional if and only if is a characteristic vector of.
  4. Let be the two-dimensional vector space and let be the linear operator on represented by the matrix relative to the standard ordered basis of. Let. Then. Therefore and so. Thus is a cyclic vector for.

    Companion matrix

Let be a linear transformation of a -dimensional vector space over a field and be a cyclic vector for. Then the vectors
form an ordered basis for. Let the characteristic polynomial for be
Then
Therefore, relative to the ordered basis, the operator is represented by the matrix
This matrix is called the companion matrix of the polynomial.