# minor (of a matrix)

Given an $n\times m$ matrix $A$ with entries ${a}_{ij}$, a *minor* of $A$ is
the determinant^{} of a smaller matrix formed from its entries by selecting
only some of the rows and columns. Let $K=\{{k}_{1},{k}_{2},\mathrm{\dots},{k}_{p}\}$ and
$L=\{{l}_{1},{l}_{2},\mathrm{\dots},{l}_{p}\}$ be subsets of $\{1,2,\mathrm{\dots},n\}$ and
$\{1,2,\mathrm{\dots},m\}$, respectively. The indices are chosen such that
$$ and $$. The $p$-th
order minor defined by $K$ and $L$ is the following determinant

$$A\left(\begin{array}{cccc}\hfill {k}_{1}\hfill & \hfill {k}_{2}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill {k}_{p}\hfill \\ \hfill {l}_{1}\hfill & \hfill {l}_{2}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill {l}_{p}\hfill \end{array}\right)=\left|\begin{array}{cccc}\hfill {a}_{{k}_{1}{l}_{1}}\hfill & \hfill {a}_{{k}_{1}{l}_{2}}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill {a}_{{k}_{1}{l}_{p}}\hfill \\ \hfill {a}_{{k}_{2}{l}_{1}}\hfill & \hfill {a}_{{k}_{2}{l}_{2}}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill {a}_{{k}_{2}{l}_{p}}\hfill \\ \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\ddots}\hfill & \hfill \mathrm{\vdots}\hfill \\ \hfill {a}_{{k}_{p}{l}_{1}}\hfill & \hfill {a}_{{k}_{p}{l}_{2}}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill {a}_{{k}_{p}{k}_{p}}\hfill \end{array}\right|.$$ |

If $p$ exceeds either $m$ or $n$, then the minor is
automatically zero. When $p=m=n$, the minor is simply the determinant
of the matrix. If $K=L$, then the minor is called *principal*.
The word *minor* may also refer to just the matrix formed from
the selected rows and columns, not necessarily its determinant. The precise
meaning is usually clear from context.

There does not seem to be a standard notation for matrix minors. Another possible notation is ${[A]}_{K,L}$.

Some authors reserve the term *minor* for the case when only one
row and one column are removed. This use is in conjunction^{} with the
concept of a *cofactor ^{} (http://planetmath.org/LaplaceExpansion)*.

Title | minor (of a matrix) |
---|---|

Canonical name | MinorofAMatrix |

Date of creation | 2013-03-22 14:07:01 |

Last modified on | 2013-03-22 14:07:01 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 7 |

Author | CWoo (3771) |

Entry type | Definition |

Classification | msc 15A15 |

Related topic | LaplaceExpansion |

Related topic | CauchyBinetFormula |

Defines | principal minor |

Defines | cofactor |