# 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**.

### 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

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

## 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

- 1954: A.C. Aitken:
*Determinants and Matrices*(8th ed.) ... (previous) ... (next): Chapter $\text I$: Definitions and Fundamental Operations of Matrices: $3$. The Notation of Matrices - 1964: Walter Ledermann:
*Introduction to the Theory of Finite Groups*(5th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Group Concept: $\S 3$: Examples of Infinite Groups: $\text{(iv)}$ - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 29$. Matrices - 1967: John D. Dixon:
*Problems in Group Theory*... (previous) ... (next): Introduction: Notation - 1980: A.J.M. Spencer:
*Continuum Mechanics*... (next): $2.1$: Matrices: $(2.1)$ - 1982: A.O. Morris:
*Linear Algebra: An Introduction*(2nd ed.) ... (previous) ... (next): Chapter $1$: Linear Equations and Matrices: $1.2$ Elementary Row Operations on Matrices: Definition $1.1$ - 1983: K.G. Binmore:
*Calculus*... (next): $1$ Vectors and matrices: $1.1$ Matrices - 1998: Richard Kaye and Robert Wilson:
*Linear Algebra*... (next): Part $\text I$: Matrices and vector spaces: $1$ Matrices: $1.1$ Matrices - 2008: David Joyner:
*Adventures in Group Theory*(2nd ed.) ... (previous) ... (next): Chapter $2$: 'And you do addition?': $\S 2.2$: Functions on vectors: $\S 2.2.3$: $m \times n$ matrices