eigenvector

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

$$Ax=\lambda x$$

for some scalar $\lambda$, i.e. such that the image of $x$ under the transformation $A$ is a scalar 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

$$(A-{\lambda }_{i}I)x_i=0$$

to find a form which characterizes the eigenvector $x_i$ (any 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.

Title	eigenvector
Canonical name	Eigenvector
Date of creation	2013-03-22 12:11:55
Last modified on	2013-03-22 12:11:55
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	12
Author	mathcam (2727)
Entry type	Definition
Classification	msc 65F15
Classification	msc 65-00
Classification	msc 15A18
Classification	msc 15-00
Related topic	SingularValueDecomposition
Related topic	Eigenvalue
Related topic	EigenvalueProblem
Related topic	SimilarMatrix
Related topic	DiagonalizationLinearAlgebra
Defines	scalar multiple