Koebe quarter theorem

2013-07-14T09:57:29Z

134.130.180.133:

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| [[Image:Mona Lisa with eigenvector.png|none|175px]]
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| width=180 style="font-size: 85%; text-align: center; " |In this [[shear (mathematics)|shear]] transformation of the [[Mona Lisa]], the central vertical axis (red vector) is unchanged, but the diagonal vector (blue) has changed direction. Hence the red vector is said to be an '''eigenvector''' of this particular transformation and the blue vector is not.
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In [[mathematics]], an '''eigenvector''' of a [[linear transformation|transformation]] is a [[vector space|vector]] which that transformation simply multiplies by a constant factor, called the '''eigenvalue''' of that vector. Often, a transformation is completely described by its eigenvalues and eigenvectors. The '''eigenspace''' for a factor is the [[set (mathematics)|set]] of eigenvectors with that factor as eigenvalue.

In the specific case of [[linear algebra]], the ''eigenvalue problem'' is this: given an ''n'' by ''n'' matrix ''A'',what nonzero vectors ''x'' in <math>R^n</math> exist, such that ''Ax'' is a scalar multiple of ''x''?

The scalar multiple is denoted by the Greek letter ''λ'' and is called an ''eigenvalue'' of the matrix A, while ''x'' is called the ''eigenvector'' of ''A'' corresponding to ''λ''. These concepts play a major role in several branches of both [[pure mathematics|pure]] and [[applied mathematics]] — appearing prominently in [[linear algebra]], [[functional analysis]], and to a lesser extent in [[nonlinear]] situations.

It is common to prefix any natural name for the vector with ''eigen'' instead of saying ''eigenvector''. For example, ''eigenfunction'' if the eigenvector is a [[function (mathematics)|function]], ''eigenmode'' if the eigenvector is a [[harmonic mode]], ''eigenstate'' if the eigenvector is a [[quantum state]], and so on. Similarly for the eigenvalue, e.g. ''eigenfrequency'' if the eigenvalue is (or determines) a [[frequency]].
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|align=left|'''[[Portal:Algebra/Selected article|...Archive]]'''
|align=center| <small>Image credit: [[User:Voyajer]]</small>
|align=right|'''[[Eigenvalue, eigenvector and eigenspace|Read more...]]'''
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Koebe quarter theorem