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.

Submatrix of order $\left({m - 1}\right) \times \left({n - 1}\right)$

Let a submatrix $\mathbf B$ of $\mathbf A$ be of order $\left({m - 1}\right) \times \left({n - 1}\right)$.

Then it is usual to denote $\mathbf B$ by indicating the (single) row and column of $\mathbf A$ which has been removed, as follows:

Let:

$a_j$ be the row of $\mathbf A$ which is not included in $\mathbf B$

$b_k$ be the column of $\mathbf A$ which is not included in $\mathbf B$.

Then the submatrix $\mathbf B$ formed from the remaining rows and columns of $\mathbf A$ can be denoted as:

$\mathbf A \left({a_j; b_k}\right)$