Cyclic vector

A cyclic vector has iterates under an operator that span the whole vector space.

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- Cyclic subspace Cyclic vector

Cyclic subspace ( $Z_{T} (x) = span {T^{j} x : j \geq 0}$ )

A cyclic subspace generated by

x

under a linear operator

T

is the span of its successive iterates

x, T x, T^{2} x, \dots

. In finite dimension it is already generated by the first

n

iterates. The first linear dependence expresses the next iterate in previous ones and proves invariance without using the Cayley-Hamilton theorem.

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Cyclic vector by

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In the context of linear algebra and vector spaces, a cyclic vector is a vector that generates a cyclic subspace under the action of a linear operator.

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