Cyclic subspace explained

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

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), T ^{2(v),} \ldots, T ^{r(v),} \ldots\} . 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 T Z(v;T) \subset 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 .

1. If

I is the identity operator then every I -cyclic subspace is one-dimensional. Z(v;T) is one-dimensional if and only if v is a characteristic vector (eigenvector) of T .

1. Let

V be the two-dimensional vector space and let T be the linear operator on V represented by the matrix \begin{bmatrix} 0 & 1 \ 0 & 0 \end{bmatrix} relative to the standard ordered basis of V . Let v = \begin{bmatrix} 0 \ 1 \end{bmatrix} . Then Tv = \begin{bmatrix} 1 \ 0 \end{bmatrix}, T ^{2v=0,} \ldots, T ^{rv=0,} \ldots . Therefore \{ v, T(v), T ^{2(v),} \ldots, T ^{r(v),} \ldots\} = \left\{ \begin{bmatrix} 0 \ 1 \end{bmatrix}, \begin{bmatrix} 1 \ 0 \end{bmatrix} \right\} 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 = \{v _{1=v,} v _{2=Tv,} v ^{2 v,} _{3= T} \ldots v _{n} = T ^{n-1} v\}

form an ordered basis for

V . Let the characteristic polynomial for T be p(x) = c _{0+c} _{1x+c} ^{2+ …} ^{2 x} + c _{n-1} x ^{n-1} + x ^{n} .

Then

\begin{align} Tv _{1} & = v _{2\\ Tv} _{2} & = v _{3\\ Tv} _{3} & = v _{4\\ \vdots} & \\ Tv _{n-1} & = v _{n\\ Tv} _{n} & = - c _{0v} _{1} - c _{1v} _{2} - … c _{n-1} v _{n \end{align}}

Therefore, relative to the ordered basis

B , the operator T is represented by the matrix \begin{bmatrix} 0 & 0 & 0 & … & 0 & - c _{0} \\ 1 & 0 & 0 & \ldots & 0 & - c _{1} \\ 0 & 1 & 0 & \ldots & 0 & - c _{2} \\ \vdots & & & & & \\ 0 & 0 & 0 & \ldots & 1 & - c _{n-1} \end{bmatrix}

This matrix is called the companion matrix of the polynomial

p(x) .^[1]

See also

• Companion matrix
• Krylov subspace

External links

• PlanetMath: cyclic subspace

Notes and References

1. Book: Hoffman . Kenneth . Kunze . Ray . Ray Kunze . 2nd . Englewood Cliffs, N.J. . 0276251 . 227 . Prentice-Hall, Inc. . Linear algebra . registration . 1971. 9780135367971 .