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 known as
Some older sources use the term array instead of matrix, but see above: the usual convention nowadays is to reserve the term array for the written-down denotation of a matrix.

The notation $\mathbf A = \sqbrk a_{m n}$ is a notation which is not yet seen frequently. $\mathbf A = \paren {a_{i j} }_{m \times n}$ or $\mathbf A = \paren {a_{i j} }$ are more common. However, the notation $\sqbrk a_{m n}$ is gaining in popularity because it better encapsulates the actual dimensions of the matrix itself in the notational form.

Some use the similar notation $\sqbrk {a_{m n} }$, moving the subscripts into the brackets.

Some sources use round brackets to encompass the array, thus:


 * $\mathbf A = \begin{pmatrix}

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{pmatrix}$

Which is used is ultimately no more than a matter of taste.

When writing a row matrix or column matrix as an array, the index of the row (for the row matrix) or column (for the column matrix) are usually left out, as the implicit $1$ is taken as understood.

Also see

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