Koebe quarter theorem: Difference between revisions
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{{Portal:Algebra/box-header|Selected Article|Portal:Algebra/Selected article/1}}</noinclude> | |||
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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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Revision as of 11:57, 14 July 2013
Template:Portal:Algebra/box-header
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| In this 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. |
In mathematics, an eigenvector of a transformation is a 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 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 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 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, 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.
| ...Archive | Image credit: User:Voyajer | Read more... |