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.
Definition.
Let formula_1 be a linear transformation of a vector space formula_2 and let formula_3 be a vector in formula_2. The formula_5-cyclic subspace of formula_2 generated by formula_7 is the subspace formula_8 of formula_2 generated by the set of vectors formula_10. This subspace is denoted by formula_11. In the case when formula_2 is a topological vector space, formula_7 is called a cyclic vector for formula_5 if formula_15 is dense in formula_2. For the particular case of finite-dimensional spaces, this is equivalent to saying that formula_15 is the whole space formula_2.
There is another equivalent definition of cyclic spaces. Let formula_1 be a linear transformation of a topological vector space over a field formula_20 and formula_7 be a vector in formula_2. The set of all vectors of the form formula_23, where formula_24 is a polynomial in the ring formula_25 of all polynomials in formula_26 over formula_20, is the formula_5-cyclic subspace generated by formula_7.
The subspace formula_15 is an invariant subspace for formula_5, in the sense that formula_32.
Companion matrix.
Let formula_54 be a linear transformation of a formula_55-dimensional vector space formula_2 over a field formula_20 and formula_7 be a cyclic vector for formula_5. Then the vectors 
form an ordered basis for formula_2. Let the characteristic polynomial for formula_5 be 
Then 
Therefore, relative to the ordered basis formula_65, the operator formula_5 is represented by the matrix 
This matrix is called the "companion matrix" of the polynomial formula_68.