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''of an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035180/e0351801.png" /> acting on a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035180/e0351802.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035180/e0351803.png" />''
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''of an operator $A$ acting on a vector space $V$ over a field $k$''
  
A non-zero vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035180/e0351804.png" /> which is mapped by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035180/e0351805.png" /> to a vector proportional to it, that is
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A non-zero vector $x \in V$ which is mapped by $A$ to a vector proportional to it, that is
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$$
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Ax = \lambda x\,,\ \ \ \lambda \in k \ .
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$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035180/e0351806.png" /></td> </tr></table>
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The coefficient $\lambda$ is called an ''[[Eigen value|eigen value]]'' of $A$.
  
The coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035180/e0351807.png" /> is called an [[Eigen value|eigen value]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035180/e0351808.png" />.
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If $A$ is a linear operator, then the set $V_\lambda$ of all eigen vectors corresponding to an eigen value $\lambda$, together with the zero vector, forms a linear subspace. It is called the eigen space of $A$ corresponding to the eigen value $\lambda$ and it coincides with the kernel $\ker(A-\lambda I)$ of the operator $A-\lambda I$ (that is, with the set of vectors mapped to 0 by this operator).  
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035180/e0351809.png" /> is a linear operator, then the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035180/e03518010.png" /> of all eigen vectors corresponding to an eigen value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035180/e03518011.png" />, together with the zero vector, forms a linear subspace. It is called the eigen space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035180/e03518012.png" /> corresponding to the eigen value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035180/e03518013.png" /> and it coincides with the kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035180/e03518014.png" /> of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035180/e03518015.png" /> (that is, with the set of vectors mapped to 0 by this operator). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035180/e03518016.png" /> is a topological vector space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035180/e03518017.png" /> a continuous operator, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035180/e03518018.png" /> is closed for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035180/e03518019.png" />. Eigen spaces need not, in general, be finite-dimensional, but if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035180/e03518020.png" /> is completely continuous (compact), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035180/e03518021.png" /> is finite-dimensional for any non-zero <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035180/e03518022.png" />.
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If $V$ is a topological vector space and $A$ a continuous operator, then $V_\lambda$ is closed for any$ \lambda \in k$. Eigen spaces need not, in general, be finite-dimensional, but if $A$ is completely continuous (compact), then $V_\lambda$ is finite-dimensional for any non-zero $\lambda$.
  
 
In fact, the existence of an eigen vector for operators on infinite-dimensional spaces is a fairly rare occurrence, although operators of special classes which are important in applications (such as integral and differential operators) often have large families of eigen vectors.
 
In fact, the existence of an eigen vector for operators on infinite-dimensional spaces is a fairly rare occurrence, although operators of special classes which are important in applications (such as integral and differential operators) often have large families of eigen vectors.
  
Generalizations of the concepts of an eigen vector and an eigen space are those of a [[Root vector|root vector]] and a root subspace. In the case of normal operators on a Hilbert space (in particular, self-adjoint or unitary operators), every root vector is an eigen vector and the eigen spaces corresponding to different eigen values are mutually orthogonal.
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Generalizations of the concepts of an eigen vector and an eigen space are those of a [[root vector]] and a root subspace. In the case of normal operators on a Hilbert space (in particular, self-adjoint or unitary operators), every root vector is an eigen vector and the eigen spaces corresponding to different eigen values are mutually orthogonal.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  K. Yosida,  "Functional analysis" , Springer  (1980)  pp. Chapt. 8, §1</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.A. [L.A. Lyusternik] Lusternik,  "Elements of functional analysis" , Hindushtan Publ. Comp.  (1974)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.V. Kantorovich,  G.P. Akilov,  "Functional analysis" , Pergamon  (1982)  pp. Chapt. 13, §3  (Translated from Russian)</TD></TR></table>
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<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  K. Yosida,  "Functional analysis" , Springer  (1980)  pp. Chapt. 8, §1</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  L.A. [L.A. Lyusternik] Lusternik,  "Elements of functional analysis" , Hindustan Publ. Comp.  (1974)  (Translated from Russian)</TD></TR>
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<TR><TD valign="top">[3]</TD> <TD valign="top">  L.V. Kantorovich,  G.P. Akilov,  "Functional analysis" , Pergamon  (1982)  pp. Chapt. 13, §3  (Translated from Russian)</TD></TR>
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</table>
  
  
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Eigen vectors are sometimes also called characteristic vectors, eigen elements, eigen functions, or proper vectors; root vectors are usually called principal vectors in the Western literature. [[#References|[a1]]] and [[#References|[a2]]] are good general Western references.
 
Eigen vectors are sometimes also called characteristic vectors, eigen elements, eigen functions, or proper vectors; root vectors are usually called principal vectors in the Western literature. [[#References|[a1]]] and [[#References|[a2]]] are good general Western references.
  
Various notions of generalized eigen vectors (or improper eigen functions) exist in the literature; e.g. see [[#References|[a3]]] and [[#References|[a4]]] for generalized eigen vectors and eigen function expansions in the context of rigged Hilbert spaces (Gel'fand triplets; see also [[Rigged Hilbert space|Rigged Hilbert space]]).
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Various notions of generalized eigen vectors (or improper eigen functions) exist in the literature; e.g. see [[#References|[a3]]] and [[#References|[a4]]] for generalized eigen vectors and eigen function expansions in the context of rigged Hilbert spaces (Gel'fand triplets; see also [[Rigged Hilbert space]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Dunford,  J.T. Schwartz,  "Linear operators. Spectral theory" , '''2''' , Interscience  (1963)  pp. Chapt. 10, §3</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.E. Taylor,  D.C. Lay,  "Introduction to functional analysis" , Wiley  (1980)  pp. Chapt. 5</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  I.M. Gel'fand,  N.Ya. Vilenkin,  "Generalized functions. Applications of harmonic analysis" , '''4''' , Acad. Press  (1964)  pp. Chapt. 1, §4  (Translated from Russian)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  Yu.M. [Yu.M. Berezanskii] Berezanskiy,  "Expansion in eigenfunctions of selfadjoint operators" , Amer. Math. Soc.  (1968)  pp. Chapt. 5, §2  (Translated from Russian)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  S. Lang,  "Linear algebra" , Addison-Wesley  (1973)</TD></TR></table>
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<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Dunford,  J.T. Schwartz,  "Linear operators. Spectral theory" , '''2''' , Interscience  (1963)  pp. Chapt. 10, §3</TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top">  A.E. Taylor,  D.C. Lay,  "Introduction to functional analysis" , Wiley  (1980)  pp. Chapt. 5</TD></TR>
 +
<TR><TD valign="top">[a3]</TD> <TD valign="top">  I.M. Gel'fand,  N.Ya. Vilenkin,  "Generalized functions. Applications of harmonic analysis" , '''4''' , Acad. Press  (1964)  pp. Chapt. 1, §4  (Translated from Russian)</TD></TR>
 +
<TR><TD valign="top">[a4]</TD> <TD valign="top">  Yu.M. [Yu.M. Berezanskii] Berezanskiy,  "Expansion in eigenfunctions of selfadjoint operators" , Amer. Math. Soc.  (1968)  pp. Chapt. 5, §2  (Translated from Russian)</TD></TR>
 +
<TR><TD valign="top">[a5]</TD> <TD valign="top">  S. Lang,  "Linear algebra" , Addison-Wesley  (1973)</TD></TR>
 +
</table>
 +
 
 +
{{TEX|done}}

Latest revision as of 13:53, 3 March 2018

of an operator $A$ acting on a vector space $V$ over a field $k$

A non-zero vector $x \in V$ which is mapped by $A$ to a vector proportional to it, that is $$ Ax = \lambda x\,,\ \ \ \lambda \in k \ . $$

The coefficient $\lambda$ is called an eigen value of $A$.

If $A$ is a linear operator, then the set $V_\lambda$ of all eigen vectors corresponding to an eigen value $\lambda$, together with the zero vector, forms a linear subspace. It is called the eigen space of $A$ corresponding to the eigen value $\lambda$ and it coincides with the kernel $\ker(A-\lambda I)$ of the operator $A-\lambda I$ (that is, with the set of vectors mapped to 0 by this operator).

If $V$ is a topological vector space and $A$ a continuous operator, then $V_\lambda$ is closed for any$ \lambda \in k$. Eigen spaces need not, in general, be finite-dimensional, but if $A$ is completely continuous (compact), then $V_\lambda$ is finite-dimensional for any non-zero $\lambda$.

In fact, the existence of an eigen vector for operators on infinite-dimensional spaces is a fairly rare occurrence, although operators of special classes which are important in applications (such as integral and differential operators) often have large families of eigen vectors.

Generalizations of the concepts of an eigen vector and an eigen space are those of a root vector and a root subspace. In the case of normal operators on a Hilbert space (in particular, self-adjoint or unitary operators), every root vector is an eigen vector and the eigen spaces corresponding to different eigen values are mutually orthogonal.

References

[1] K. Yosida, "Functional analysis" , Springer (1980) pp. Chapt. 8, §1
[2] L.A. [L.A. Lyusternik] Lusternik, "Elements of functional analysis" , Hindustan Publ. Comp. (1974) (Translated from Russian)
[3] L.V. Kantorovich, G.P. Akilov, "Functional analysis" , Pergamon (1982) pp. Chapt. 13, §3 (Translated from Russian)


Comments

It is also quite common to write eigenvector, eigenspace, etc., i.e. not two words but one.

Eigen vectors are sometimes also called characteristic vectors, eigen elements, eigen functions, or proper vectors; root vectors are usually called principal vectors in the Western literature. [a1] and [a2] are good general Western references.

Various notions of generalized eigen vectors (or improper eigen functions) exist in the literature; e.g. see [a3] and [a4] for generalized eigen vectors and eigen function expansions in the context of rigged Hilbert spaces (Gel'fand triplets; see also Rigged Hilbert space).

References

[a1] N. Dunford, J.T. Schwartz, "Linear operators. Spectral theory" , 2 , Interscience (1963) pp. Chapt. 10, §3
[a2] A.E. Taylor, D.C. Lay, "Introduction to functional analysis" , Wiley (1980) pp. Chapt. 5
[a3] I.M. Gel'fand, N.Ya. Vilenkin, "Generalized functions. Applications of harmonic analysis" , 4 , Acad. Press (1964) pp. Chapt. 1, §4 (Translated from Russian)
[a4] Yu.M. [Yu.M. Berezanskii] Berezanskiy, "Expansion in eigenfunctions of selfadjoint operators" , Amer. Math. Soc. (1968) pp. Chapt. 5, §2 (Translated from Russian)
[a5] S. Lang, "Linear algebra" , Addison-Wesley (1973)
How to Cite This Entry:
Eigen vector. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Eigen_vector&oldid=13516
This article was adapted from an original article by T.S. PigolkinaV.S. Shul'man (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article