In mathematics, in linear algebra, a cyclic subspace is a certain special subspace of a finite-dimensional 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 . If , then is called a cyclic vector for .
There is another equivalent definition of cyclic spaces. Let be a linear transformation of a finite dimensional 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 .
- For any vector space and any linear operator on , the -cyclic subspace generated by the zero vector is the zero-subspace of .
- If is the identity operator then every -cyclic subspace is one-dimensional.
- is one-dimensional if and only if is a characteristic vector (eigenvector) of .
- 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 .
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 .
- PlanetMath: cyclic subspace