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.


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  . [1]

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  .[1]

The subspace   is an invariant subspace for  , in the sense that  .


  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 (eigenvector) 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 matrixEdit

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




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


This matrix is called the companion matrix of the polynomial  .[1]

See alsoEdit

External linksEdit


  1. ^ a b c Hoffman, Kenneth; Kunze, Ray (1971). Linear algebra (2nd ed.). Englewood Cliffs, N.J.: Prentice-Hall, Inc. p. 227. MR 0276251.