# Definition:Array

## Definition

An **array** is an arrangement (usually rectangular) of objects (usually numbers) such that the relations between the entries borne by their relative positions in that arrangement have a particular significance.

### Dimensions of Array

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

Then the parameters $m$ and $n$ are known as the **dimensions** of the array.

### Element of Array

Let $\mathbf A$ be an Array.

The individual $n \times n$ symbols that go to form $\mathbf L$ are known as the **elements** of $\mathbf L$.

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 $a \left({i, j}\right)$ can be used.

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

### Row of Array

Let $\mathbf A$ be an array.

The **rows** of $\mathbf A$ are the lines of elements reading **across** the page.

### Column of Array

Let $\mathbf A$ be an array.

The **columns** of $\mathbf A$ are the lines of elements reading **down** the page.

### Diagonal of Array

Let $\mathbf A = \left[{a}\right]_{m n}$ be an array.

The **diagonals** are the lines of elements of $\mathbf A$ running from:

- $(1): \quad$ the element in the first row and first column running downwards and to the right
- $(2): \quad$ the element in the first row and last column running downwards and to the left
- $(3): \quad$ the element in the last row and first column running upwards and to the right
- $(4): \quad$ the element in the last row and last column running upwards and to the left

If $m = n$, that is, if $\mathbf A$ is a square array, then $(1)$ and $(4)$ coincide, and $(2)$ and $(3)$ also coincide.

## Square Array

An array whose dimensions are equal is called a **square array**.

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

A **square array** $\left[{a}\right]_{n n}$ is usually denoted $\left[{a}\right]_n$.

## Examples

### Arbitrary Example

An example of a rectangular Array:

- $\begin {bmatrix} \alpha_{1 1} & \alpha_{1 2} & \cdots & \alpha_{1 n} \\ \alpha_{2 1} & \alpha_{2 2} & \cdots & \alpha_{2 n} \\ \vdots & \vdots & \ddots & \vdots \\ \alpha_{m 1} & \alpha_{m 2} & \cdots & \alpha_{m n} \end {bmatrix}$

## Also see

- Definition:Matrix
- Definition:Pascal's Triangle
- Definition:Stirling's Triangles
- Definition:Cayley Table
- Definition:Latin Square
- Definition:Truth Table
- Definition:Magic Square

- Results about
**arrays**can be found here.

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 29$. Matrices