\begin{definition}[Definition:Matrix]
Let $S$ be a set.
Let $m, n \in \Z_{>0}$ be strictly positive integers.
An '''$m \times n$ matrix over $S$''' (said '''$m$ times $n$''' or '''$m$ by $n$''') is a mapping from the cartesian product of two integer intervals $\closedint 1 m \times \closedint 1 n$ into $S$.
When the set $S$ is understood, or for the purpose of the particular argument irrelevant, we can refer just to an '''$m \times n$ matrix'''.
The convention is for the variable representing the '''matrix''' itself to be represented in $\mathbf {boldface}$.
A '''matrix''' is frequently written as a rectangular array, and when reference is being made to how it is written down, will sometimes be called an '''array'''.
For example, let $\mathbf A$ be an $m \times n$ '''matrix'''. This can be written as the following array:
:$\mathbf A = \begin {bmatrix}
a_{1 1} & a_{1 2} & \cdots & a_{1 n} \\
a_{2 1} & a_{2 2} & \cdots & a_{2 n} \\
 \vdots &  \vdots & \ddots &  \vdots \\
a_{m 1} & a_{m 2} & \cdots & a_{m n} \\
\end{bmatrix}$
Thus an $m \times n$ '''matrix''' has $m$ rows and $n$ columns.
Note that no commas are placed between elements in the rows.
It needs to be understood that, when writing a '''matrix''', it is important to leave sufficient space between the elements for the columns to be distinct.
An $m \times n$ '''matrix''' can also be written as $\mathbf A = \sqbrk a_{m n}$, where the subscripts $m$ and $n$ denote respectively the number of rows and the number of columns in the '''matrix'''.
\end{definition}