Definition:Submatrix/Notation

Definition
Let $\mathbf A$ be a matrix with $m$ rows and $n$ columns.

A submatrix of $\mathbf A$ is denoted as follows.

Let:
 * $\left\{ {a_1, a_2, \ldots, a_r}\right\}$ be the indices of the $r$ selected rows
 * $\left\{ {b_1, b_2, \ldots, b_s}\right\}$ be the indices of the $s$ selected columns

where all of $a_1, \ldots, a_r$ are between $1$ and $m$, and all of $b_1, \ldots, b_s$ are between $1$ and $n$.

Then the submatrix formed from rows $\left\{ {a_1, a_2, \ldots, a_r}\right\}$ and columns $\left\{ {b_1, b_2, \ldots, b_s}\right\}$ is denoted as:


 * $\mathbf A \left[{a_1, a_2, \ldots, a_r; b_1, b_2, \ldots, b_s}\right]$

It is usual to specify the rows and columns in ascending numerical order.