Definition:Matrix

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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.


Order

Let $\sqbrk a_{m n}$ be an $m \times n$ matrix.

Then the parameters $m$ and $n$ are known as the order of the matrix.


Element

Let $\mathbf A$ be an $m \times n$ matrix over a set $S$.

The individual $m \times n$ elements of $S$ that go to form $\mathbf A = \sqbrk a_{m n}$ are known as the elements of the matrix.

The element at row $i$ and column $j$ is called element $\tuple {i, j}$ of $\mathbf A$, and can be written $a_{i j}$, or $a_{i, j}$ if $i$ and $j$ are of more than one character.

If the indices are still more complicated coefficients and further clarity is required, then the form $a \tuple {i, j}$ can be used.


Note that the first subscript determines the row, and the second the column, of the matrix where the element is positioned.


Indices

Let $\mathbf A$ be an $m \times n$ matrix.

Let $a_{i j}$ be the element in row $i$ and column $j$ of $\mathbf A$.


Then the subscripts $i$ and $j$ are referred to as the indices (singular: index) of $a_{i j}$.


Row

Let $\mathbf A$ be an $m \times n$ matrix.

For each $i \in \closedint 1 m$, the rows of $\mathbf A$ are the ordered $n$-tuples:

$r_i = \tuple {a_{i 1}, a_{i 2}, \ldots, a_{i n} }$

where $r_i$ is called the $i$th row of $\mathbf A$.


A row of an $m \times n$ matrix can also be treated as a $1 \times n$ row matrix in its own right:

$r_i = \begin {bmatrix} a_{i 1} & a_{i 2} & \cdots & a_{i n} \end {bmatrix}$

for $i = 1, 2, \ldots, m$.


Column

Let $\mathbf A$ be an $m \times n$ matrix.

For each $j \in \closedint 1 n$, the columns of $\mathbf A$ are the ordered $m$-tuples:

$c_j = \tuple {a_{1 j}, a_{2 j}, \ldots, a_{m j} }$

where $c_j$ is called the $j$th column of $\mathbf A$.


A column of an $m \times n$ matrix can also be treated as a $m \times 1$ column matrix in its own right:

$c_j = \begin {bmatrix} a_{1 j} \\ a_{2 j} \\ \vdots \\ a_{m j} \end {bmatrix}$

for $j = 1, 2, \ldots, n$.


Underlying Structure

Let $\mathbf A$ be a matrix over a set $S$.

The set $S$ can be referred to as the underlying set of $\mathbf A$.

In the context of matrices, however, it is usual for $S$ itself to be the underlying set of an algebraic structure in its own right.

If this is the case, then the structure $\struct {S, \circ_1, \circ_2, \ldots, \circ_n}$ (which may also be an ordered structure) can be referred to as the underlying structure of $\mathbf A$.


When the underlying structure is not specified, it is taken for granted that it is one of the standard number systems, usually the real numbers $\R$.


Square Matrix

An $n \times n$ matrix is called a square matrix.

That is, a square matrix is a matrix which has the same number of rows as it has columns.

A square matrix is usually denoted $\sqbrk a_n$ in preference to $\sqbrk a_{n n}$.


In contrast, a non-square matrix can be referred to as a rectangular matrix.


Diagonal Elements

Let $\mathbf A = \sqbrk a_{m n}$ be a matrix.

The elements $a_{j j}: j \in \closedint 1 {\min \set {m, n} }$ constitute the main diagonal of the matrix.

The elements themselves are called the diagonal elements.


Lower Triangular Elements

Let $\mathbf A = \sqbrk a_{m n}$ be a matrix.

The elements $a_{i j}: i > j$ are called the lower triangular elements.


Upper Triangular Elements

Let $\mathbf A = \sqbrk a_{m n}$ be a matrix.

The elements $a_{i j}: i < j$ are called the upper triangular elements.


Zero Row or Column

Let $\mathbf A = \sqbrk a_{m n}$ be an $m \times n$ matrix whose underlying structure is a ring or field (usually numbers).


If a row or column of $\mathbf A$ contains only zeroes, then it is a zero row or a zero column.


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.


Einstein Summation Convention

The Einstein summation convention is a notational device used in the manipulation of matrices and vectors, in particular square matrices in the context of physics and applied mathematics.

If the same index occurs twice in a given expression involving matrices, then summation over that index is automatically assumed.

Thus the summation sign can be omitted, and expressions can be written more compactly.


Examples

Example of a $3 \times 4$ Matrix

An example of a matrix of order $3 \times 4$ is:

$\mathbf A := \begin {bmatrix} 1 & 0 & -3 & 1 \\ 2 & 1 & 3 & 1 \\ 1 & 0 & 1 & 1 \end {bmatrix}$

Row $2$ of $\mathbf A$ is $\begin {bmatrix} 2 & 1 & 3 & 1 \end {bmatrix}$.

Column $3$ of $\mathbf A$ is $\begin {bmatrix} -3 \\ 3 \\ 1 \end {bmatrix}$.


$1 \times 1$ Matrix

A matrix of order $1 \times 1$ is a single element:

$\mathbf B := \begin {bmatrix} b \end {bmatrix}$

Such a matrix can be identified with a scalar, that is: an element of the underlying set


Also see


Historical Note

The concept of a matrix is generally considered to have originated with Arthur Cayley.

Cayley was the first to regard a matrix as an operator on a tuple of variables.


Linguistic Note

The plural form of matrix is matrices, pronounced may-tri-seez.


Compare with index (plural indices), apex (plural apices), and so on.


Sources

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