# 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