View source for Eigenvalues and eigenvectors

{{complex|date=April 2012}}
[[File:Mona Lisa with eigenvector.png|thumb|270px|Illustration of a transformation (of [[Mona Lisa]]): The image is changed in such a way that the red arrow (vector) does not change its direction, but the blue one does. The red vector therefore is an eigenvector of this transformation, the blue one is not. Since the red vector does not change its length, its eigenvalue is 1. The transformation used is called [[shear mapping]].]]

[[Linear algebra]] talks about [[function (mathematics)|functions]], which are often called transformations. In that context, an '''eigenvector''' is a [[vector]]—different from the [[null vector]]—which does not change direction in the transformation (except if it turns the vector exactly around). The vector may change its length, or become null. The value of the change in length of the vector is known as '''eigenvalue'''.

==Basics==
If there exists a '''[[square matrix]]''' called '''''A''''', a '''[[scalar]] ''λ''''', and a '''non-zero vector''' '''v''', then '''''λ''''' '''is the Eigenvalue''' and '''v is the Eigenvector''' if the following equation is satisfied:

:<math>A\mathbf{v} = \lambda \mathbf{v} \, .</math>

In other words, if matrix ''A'' times the vector '''v''' is equal to the scalar ''λ'' times the vector '''v''', then ''λ'' is the eigenvalue of '''v''', where '''v''' is the eigenvector.

An '''eigenspace''' of ''A'' is the set of all eigenvectors with the same eigenvalue together with the [[zero vector]]. However, the zero vector is not an eigenvector.<ref>{{Cite web
| title=Eigenvector
| publisher=Wolfram Research, Inc.
| url=http://mathworld.wolfram.com/Eigenvector.html
| accessdate = 29 January 2010
}}</ref>

These ideas often are extended to more general situations, where scalars are elements of any [[field (mathematics)|field]], vectors are elements of any vector space, and [[linear transformations]] may or may not be represented by matrix multiplication. For example, instead of [[real numbers]], scalars may be [[complex numbers]]; instead of arrows, vectors may be  [[function (mathematics)|functions]] or [[frequency|frequencies]]; instead of matrix multiplication, linear transformations may be [[operator (mathematics)|operators]] such as the [[Derivative (mathematics)|derivative]] from [[calculus]]. These are only a few of countless examples where eigenvectors and eigenvalues are important.

In cases like these, the idea of ''direction'' loses its ordinary meaning, and has a more abstract definition instead. But even in this case, if that abstract ''direction'' is unchanged by a given linear transformation, the prefix "eigen" is used, as in ''[[eigenfunction]]'', ''[[eigenmode]]'', ''[[eigenface]]'', ''eigenstate'', and ''eigenfrequency''.

Eigenvalues and eigenvectors have many applications in both [[mathematics|pure]] and [[applied mathematics]]. They are used in [[matrix factorization]], in [[quantum mechanics]], [[facial recognition system]]s, and in many other areas.

==Example==
For the matrix ''A''
:<math>A = \begin{bmatrix} 2 & 1\\1 & 2 \end{bmatrix}.</math>
the vector 
:<math>\mathbf x = \begin{bmatrix} 3 \\ -3 \end{bmatrix}</math>
is an eigenvector with eigenvalue 1. Indeed,
:<math>A \mathbf x = \begin{bmatrix} 2 & 1\\1 & 2 \end{bmatrix} \begin{bmatrix} 3 \\ -3 \end{bmatrix} = \begin{bmatrix} (2 \cdot 3) + (1 \cdot (-3)) \\ (1 \cdot 3) + (2 \cdot (-3)) \end{bmatrix} = \begin{bmatrix} 3 \\ -3 \end{bmatrix} = 1 \cdot \begin{bmatrix} 3 \\ -3 \end{bmatrix}.</math>
On the other hand the vector 
:<math>\mathbf x = \begin{bmatrix} 0 \\ 1 \end{bmatrix}</math>
is ''not'' an eigenvector, since
:<math>\begin{bmatrix} 2 & 1\\1 & 2 \end{bmatrix} \begin{bmatrix} 0 \\ 1 \end{bmatrix} = \begin{bmatrix} (2 \cdot 0) + (1 \cdot 1) \\ (1 \cdot 0) + (2 \cdot 1) \end{bmatrix} = \begin{bmatrix} 1 \\ 2 \end{bmatrix}.</math>
and this vector is not a multiple of the original vector '''x'''.

<!--
==Other pages==
* [[Nonlinear eigenproblem]]
* [[Quadratic eigenvalue problem]]
* [[Introduction to eigenstates]]
* [[Eigenplane]]
* [[Jordan normal form]]
* [[List of numerical analysis software]]
-->

==Notes==
{{reflist}}

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== Other websites ==
{{Wikibooks|Linear Algebra|Eigenvalues and Eigenvectors}}
{{Wikibooks|The Book of Mathematical Proofs|Algebra/Linear Transformations}}
* [http://www.physlink.com/education/AskExperts/ae520.cfm What are Eigen Values?] — non-technical introduction from PhysLink.com's "Ask the Experts"
*[http://khanexercises.appspot.com/video?v=PhfbEr2btGQ Introduction to Eigen Vectors and Eigen Values] – lecture from Kahn Academy

;Theory
* [http://mathworld.wolfram.com/Eigenvector.html Eigenvector] — Wolfram MathWorld
* [http://ocw.mit.edu/ans7870/18/18.06/javademo/Eigen/ Eigen Vector Examination working applet]
* [http://web.mit.edu/18.06/www/Demos/eigen-applet-all/eigen_sound_all.html Same Eigen Vector Examination as above in a Flash demo with sound]
* [http://www.sosmath.com/matrix/eigen1/eigen1.html Computation of Eigenvalues]
* [http://www.cs.utk.edu/~dongarra/etemplates/index.html Numerical solution of eigenvalue problems], edited by Zhaojun Bai, James Demmel, Jack Dongarra, Axel Ruhe and Henk van der Vorst

;Online calculators
* [http://www.arndt-bruenner.de/mathe/scripts/engl_eigenwert.htm arndt-bruenner.de]
* [http://www.bluebit.gr/matrix-calculator/ bluebit.gr]
* [http://wims.unice.fr/wims/wims.cgi?session=6S051ABAFA.2&+lang=en&+module=tool%2Flinear%2Fmatrix.en wims.unice.fr]

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