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.

## Contents

* 1 Definition
* 1.1 Examples
* 2 Companion matrix
* 3 See also
* 4 External links
* 5 References

## 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, denoted Z(v;T), is the subspace of V generated by the set of vectors {v,T(v),T2(v),…,Tr(v),…}. In the case when V is a topological vector space, v is called a cyclic vector for T if Z(v;T) is dense in V. For the particular case of finite-dimensional spaces, this is equivalent to saying that Z(v;T) is the whole space V. [1]

There is another equivalent definition of cyclic spaces. Let T:V->V be a linear transformation of a topological 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.[1]

The subspace Z(v;T) is an invariant subspace for T, in the sense that TZ(v;T)⊂Z(v;T).

### 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 [0100] relative to the standard ordered basis of V. Let v=[01]. Then Tv=[10],T2v=0,…,Trv=0,…. Therefore {v,T(v),T2(v),…,Tr(v),…}={[01],[10]} 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

B={v1=v,v2=Tv,v3=T2v,…vn=Tn−1v}

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

p(x)=c0+c1x+c2x2+⋯+cn−1xn−1+xn.

Then

Tv1=v2Tv2=v3Tv3=v4⋮Tvn−1=vnTvn=−c0v1−c1v2−⋯cn−1vn

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

[000⋯0−c0100…0−c1010…0−c2⋮000…1−cn−1]

This matrix is called the companion matrix of the polynomial p(x).[1]

## See also

* Companion matrix
* Krylov subspace

## External links

* PlanetMath: cyclic subspace

## References

1. ↑ 1.0 1.1 1.2 Hoffman, Kenneth (1971). Linear algebra (2nd ed.). Englewood Cliffs, N.J.: Prentice-Hall, Inc.. p. 227. ISBN 9780135367971. https://archive.org/details/linearalgebra00hoff_0.

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