View source for Cyclic vector

{{TEX|done}}{{MSC|15A|47A16,93B}}

Let $A$ be an endomorphism of a finite-dimensional [[Vector space|vector space]] $V$. A cyclic vector for $A$ is a vector $v$ such that $v,Av,\dots,A^{n-1}v$ form a basis for $V$, ''i.e.'' such that the pair $(A,v)$ is completely reachable (see also [[Pole assignment problem|Pole assignment problem]]; [[Majorization ordering|Majorization ordering]]; [[System of subvarieties|System of subvarieties]]; [[Frobenius matrix|Frobenius matrix]]).

A vector $v$ in an (infinite-dimensional) [[Banach space|Banach space]] or [[Hilbert space|Hilbert space]] with an operator $A$ on it is said to be cyclic if the linear combinations of the vectors $A^iv$, $i=0,1,\dots$, form a dense subspace, [[#References|[a1]]].

More generally, let $\mathcal A$ be a subalgebra of $\mathcal B(H)$, the algebra of bounded operators on a Hilbert space $H$. Then $v\in H$ is cyclic if $\mathcal Av$ is dense in $H$, [[#References|[a2]]], [[#References|[a5]]].

If $\phi$ is a [[Unitary representation|unitary representation]] of a (locally compact) group $G$ in $H$, then $v\in H$ is called cyclic if the linear combinations of the $\phi(g)v$, $g\in G$, form a dense set, [[#References|[a3]]], [[#References|[a4]]]. For the connection between positive-definite functions on $G$ and the cyclic representations (i.e., representations that admit a cyclic vector), see [[Positive-definite function on a group|Positive-definite function on a group]]. An [[Irreducible representation|irreducible representation]] is cyclic with respect to every non-zero vector.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Reed,   B. Simon,   "Methods of mathematical physics: Functional analysis" , '''1''' , Acad. Press  (1972)  pp. 226ff</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  R.V. Kadison,   J.R. Ringrose,   "Fundamentals of the theory of operator algebras" , '''1''' , Acad. Press  (1983)  pp. 276</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  S.A. Gaal,   "Linear analysis and representation theory" , Springer  (1973)  pp. 156</TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top">  A.A. Kirillov,   "Elements of the theory of representations" , Springer  (1976)  pp. 53  (In Russian)</TD></TR>
<TR><TD valign="top">[a5]</TD> <TD valign="top">  M.A. Naimark,   "Normed rings" , Noordhoff  (1964)  pp. 239  (In Russian) {{ZBL|0137.31703}}</TD></TR>
</table>