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# 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 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

$(A-\lambda_{i}I)x_{i}=0$ |

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.

Defines:

scalar multiple

Related:

SingularValueDecomposition, Eigenvalue, EigenvalueProblem, SimilarMatrix, DiagonalizationLinearAlgebra

Type of Math Object:

Definition

Major Section:

Reference

## Mathematics Subject Classification

65F15*no label found*65-00

*no label found*15A18

*no label found*15-00

*no label found*

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