Definition:Matrix

Definition
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.

Also defined as
Some (in fact most) sources in elementary mathematics define a matrix as a rectangular array of numbers.

This definition is adequate for most applications of the theory.

Also presented as
Lines may if desired be drawn between rows and columns of an array in order to clarify its sections, for example:


 * $\sqbrk {\begin {array} {ccc|cc}

a_{11} & a_{12} & a_{13} & b_{11} & b_{11} \\ a_{21} & a_{22} & a_{23} & b_{21} & b_{21} \\ \hline c_{11} & c_{12} & c_{13} & d_{11} & d_{12} \\ c_{21} & c_{22} & c_{23} & d_{21} & d_{22} \\ c_{31} & c_{32} & c_{33} & d_{31} & d_{32} \\ \end {array} }$

Also see

 * Definition:Block Matrix
 * Linear Transformation as Matrix Product
 * Matrix Product as Linear Transformation