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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120320/c1203201.png" /> be an endomorphism of a finite-dimensional [[Vector space|vector space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120320/c1203202.png" />. A cyclic vector for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120320/c1203203.png" /> is a vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120320/c1203204.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120320/c1203205.png" /> form a basis for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120320/c1203206.png" />, i.e. such that the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120320/c1203207.png" /> 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]]).
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120320/c1203208.png" /> in an (infinite-dimensional) [[Banach space|Banach space]] or [[Hilbert space|Hilbert space]] with an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120320/c1203209.png" /> on it is said to be cyclic if the linear combinations of the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120320/c12032010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120320/c12032011.png" />, form a dense subspace, [[#References|[a1]]].
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120320/c12032012.png" /> be a subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120320/c12032013.png" />, the algebra of bounded operators on a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120320/c12032014.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120320/c12032015.png" /> is cyclic if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120320/c12032016.png" /> is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120320/c12032017.png" />, [[#References|[a2]]], [[#References|[a5]]].
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120320/c12032018.png" /> is a [[Unitary representation|unitary representation]] of a (locally compact) group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120320/c12032019.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120320/c12032020.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120320/c12032021.png" /> is called cyclic if the linear combinations of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120320/c12032022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120320/c12032023.png" />, form a dense set, [[#References|[a3]]], [[#References|[a4]]]. For the connection between positive-definite functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120320/c12032024.png" /> 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.
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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====
 
====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)</TD></TR></table>
 
<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)</TD></TR></table>

Revision as of 16:04, 20 September 2014

Let $A$ be an endomorphism of a finite-dimensional 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; Majorization ordering; System of subvarieties; Frobenius matrix).

A vector $v$ in an (infinite-dimensional) Banach space or 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, [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 $H$, [a2], [a5].

If $\phi$ is a 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, [a3], [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. An irreducible representation is cyclic with respect to every non-zero vector.

References

[a1] M. Reed, B. Simon, "Methods of mathematical physics: Functional analysis" , 1 , Acad. Press (1972) pp. 226ff
[a2] R.V. Kadison, J.R. Ringrose, "Fundamentals of the theory of operator algebras" , 1 , Acad. Press (1983) pp. 276
[a3] S.A. Gaal, "Linear analysis and representation theory" , Springer (1973) pp. 156
[a4] A.A. Kirillov, "Elements of the theory of representations" , Springer (1976) pp. 53 (In Russian)
[a5] M.A. Naimark, "Normed rings" , Noordhoff (1964) pp. 239 (In Russian)
How to Cite This Entry:
Cyclic vector. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cyclic_vector&oldid=33350
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article