# submatrix notation

Let $n$ and $k$ be integers with $1\le k\le n$. Denote by
${Q}_{k,n}$ the totality of all sequences of $k$ integers, where the elements of
the sequence are strictly increasing^{} and choosen from $\{1,\mathrm{\dots},n\}$.

Let $A=({a}_{ij})$ be an $m\times n$ matrix with elements from some set, usually taken to be a field for ring. Let $k$ and $r$ be positive integers with $1\le k\le m$, $1\le r\le n$, $\alpha \in {Q}_{k,m}$ and $\beta \in {Q}_{r,n}$. We let $\alpha =({i}_{1},\mathrm{\dots},{i}_{k})$ and $\beta =({j}_{1},\mathrm{\dots},{j}_{r})$

The submatrix^{} $A[\alpha ,\beta ]$ has $(s,t)$ entry equal to
${a}_{{i}_{s}{j}_{t}}$ and has $k$ rows and $r$ columns.

We denote by $A(\alpha ,\beta )$ the submatrix of $A$ whose rows and columns are complementary to $\alpha $ and $\beta $, respectively.

We can also define similarly the notations $A[\alpha ,\beta )$ and $A(\alpha ,\beta ]$.

Title | submatrix notation |
---|---|

Canonical name | SubmatrixNotation |

Date of creation | 2013-03-22 16:13:36 |

Last modified on | 2013-03-22 16:13:36 |

Owner | Mathprof (13753) |

Last modified by | Mathprof (13753) |

Numerical id | 5 |

Author | Mathprof (13753) |

Entry type | Definition |

Classification | msc 15-00 |