# 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 |