# multiplicity of eigenvalue

Suppose $V$ is a finite dimensional vector space^{} over a field $\mathbb{F}$,
and suppose $L:V\to V$ is a linear map.
Suppose also that $\lambda \in \mathbb{F}$ is an
eigenvalue^{} of $L$, that is, $\mathrm{det}(L-\lambda I)=0$.

The *algebraic multiplicity*,
denoted by ${A}_{\lambda}(L)$, of $\lambda $
is the multiplicity of the root $\lambda $ to the polynomial
$\mathrm{det}(L-\lambda I)=0$.
The *geometric multiplicity* of $\lambda $, denoted by
${G}_{\lambda}(L)$, is the
dimension^{} of $\mathrm{ker}(L-\lambda I)$, the eigenspace^{} of $\lambda $.

Title | multiplicity of eigenvalue |
---|---|

Canonical name | MultiplicityOfEigenvalue |

Date of creation | 2013-03-22 15:15:15 |

Last modified on | 2013-03-22 15:15:15 |

Owner | matte (1858) |

Last modified by | matte (1858) |

Numerical id | 5 |

Author | matte (1858) |

Entry type | Definition |

Classification | msc 15A18 |

Defines | geometric multiplicity |

Defines | algebraic multiplicity |