# Definition:Matrix

## Contents

## 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:

- $\mathbf A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \\ \end{bmatrix}$

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 pays 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**.

Arrays may also be conveniently represented on the page by placing symbols together that denote other matrices.

For example, let $\mathbf A = \sqbrk a_{m n}, \mathbf B = \sqbrk b_{m p}, \mathbf C = \sqbrk c_{r n}, \mathbf D = \sqbrk d_{r p}$.

We can create the $\paren {m + r} \times \paren {n + p}$ matrix $\mathbf M = \begin{bmatrix} \mathbf A & \mathbf B \\ \mathbf C & \mathbf D \end{bmatrix}$.

It is clear that the dimensions of the component matrices must be compatible for this construct to be defined.

Lines may if desired be drawn between rows and columns of an array in order to clarify its sections.

### Dimensions

Let $\left[{a}\right]_{m n}$ be an $m \times n$ matrix.

Then the parameters $m$ and $n$ are known as the **dimensions** 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 an element 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 $\left({S, \circ_1, \circ_2, \ldots, \circ_n}\right)$ (which may also be an ordered structure) can be referred to as the **underlying structure of $\mathbf A$**.

### 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 **principal diagonal** or **main diagonal** of the matrix, and 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 $\mathbf A = \left[{a}\right]_{m n}$ 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 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:

- $\mathbf A = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \\ \end{pmatrix}$

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.

## Summation Convention

The **summation convention** is a notational device used in the manipulation of matrices, 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.

## Also see

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

## 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

- 1964: Walter Ledermann:
*Introduction to the Theory of Finite Groups*(5th ed.) ... (previous) ... (next): $\S 3$: Examples of Infinite Groups: $\text{(iv)}$ - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 29$ - 1967: John D. Dixon:
*Problems in Group Theory*... (previous) ... (next): Introduction: Notation - 1980: A.J.M. Spencer:
*Continuum Mechanics*... (next): $2.1$: Matrices: $(2.1)$ - 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