# eigenvalue

Let $V$ be a vector space^{} over $k$ and $T$ a linear operator^{} on $V$.
An eigenvalue^{} for $T$ is an scalar $\lambda $ (that is, an element of $k$) such that
$T(z)=\lambda z$ for some nonzero vector $z\in V$.
Is that case, we also say that $z$ is an eigenvector^{} of $T$.

This can also be expressed as follows: $\lambda $ is an eigenvalue for $T$ if the kernel of $A-\lambda I$ is non trivial.

A linear operator can have several eigenvalues (but no more than the dimension^{} of the space). Eigenvectors corresponding to different eigenvalues are linearly independent^{}.

Title | eigenvalue |

Canonical name | Eigenvalue1 |

Date of creation | 2013-03-22 14:01:53 |

Last modified on | 2013-03-22 14:01:53 |

Owner | drini (3) |

Last modified by | drini (3) |

Numerical id | 8 |

Author | drini (3) |

Entry type | Definition |

Classification | msc 15A18 |

Related topic | LinearTransformation |

Related topic | Scalar |

Related topic | Vector |

Related topic | Kernel |

Related topic | Dimension2 |

Defines | eigenvector |