Cyclic subspace

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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.

Definition[edit]

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 .[1]

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 .[1]

Examples[edit]

  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 matrix[edit]

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 .[1]

See also[edit]

External links[edit]

References[edit]

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