PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (211)
Orphanage
Unclass'd (1)
Unproven (472)
Corrections (36)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: Medium Entry average rating: No information on entry rating
eigenvector (Definition)

Let $ A$ be an $ n \times n$ square matrix and $ x$ an $ n\times 1$ column vector. Then a (right) eigenvector of $ A$ is a nonzero vector $ x$ such that

$\displaystyle Ax = \lambda x $

for some scalar $ \lambda$, i.e. such that the image of $ x$ under the transformation $ A$ is a scalar multiple of $ x$. One can similarly define left eigenvectors in the case that $ A$ acts on the right.

One can find eigenvectors by first finding eigenvalues, then for each eigenvalue $ \lambda_i$, solving the system

$\displaystyle (A-\lambda_i I) x_i = 0 $

to find a form which characterizes the eigenvector $ x_i$ (any multiple of $ x_i$ is also an eigenvector). Of course, this is not necessarily the best way to do it; for this, see singular value decomposition.



"eigenvector" is owned by mathcam. [ full author list (2) | owner history (1) ]
(view preamble)

View style:

See Also: singular value decomposition, eigenvalue, eigenvalue problem, similar matrix, diagonalization

Also defines:  scalar multiple

Attachments:
generalized eigenvector (Definition) by CWoo
Log in to rate this entry.
(view current ratings)

Cross-references: singular value decomposition, eigenvalue, finding eigenvalues, acts on, eigenvectors, transformation, image, scalar, vector, right, column vector, square matrix
There are 24 references to this entry.

This is version 8 of eigenvector, born on 2002-01-19, modified 2007-05-12.
Object id is 1497, canonical name is Eigenvector.
Accessed 37881 times total.

Classification:
AMS MSC: 15-00 (Linear and multilinear algebra; matrix theory :: General reference works )
 15A18 (Linear and multilinear algebra; matrix theory :: Eigenvalues, singular values, and eigenvectors)
 65-00 (Numerical analysis :: General reference works )
 65F15 (Numerical analysis :: Numerical linear algebra :: Eigenvalues, eigenvectors)
Pending Errata and Addenda
None.
[ View all 5 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)