# algebraic and geometric multiplicity do not coincide

Zero is an eigenvalue^{} of

$$A=\left(\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right)$$ |

with algebraic multiplicity $2$ and geometric multiplicity $1$.

Indeed, as

$$det(A-\lambda I)={\lambda}^{2}$$ |

it follows that $0$ is an eigenvalue of $A$ with algebraic multiplicity $2$. To find the geometric multiplicity of $A$ we need to calculate $\mathrm{ker}A$. Thus, suppose

$$\left(\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right)\left(\begin{array}{c}\hfill a\hfill \\ \hfill b\hfill \end{array}\right)=\left(\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \end{array}\right).$$ |

This implies $b=0$, so

$$\mathrm{ker}A=\mathrm{span}\left(\begin{array}{c}\hfill 1\hfill \\ \hfill 0\hfill \end{array}\right),$$ |

and the geometric multiplicity of $0$ is $1$.

Title | algebraic and geometric multiplicity do not coincide |
---|---|

Canonical name | AlgebraicAndGeometricMultiplicityDoNotCoincide |

Date of creation | 2013-03-22 15:15:18 |

Last modified on | 2013-03-22 15:15:18 |

Owner | matte (1858) |

Last modified by | matte (1858) |

Numerical id | 5 |

Author | matte (1858) |

Entry type | Example |

Classification | msc 15A18 |