Definition:Array

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Definition

An array is an arrangement (usually rectangular) of objects of the same type (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 $\sqbrk a_{m n}$ be an $m \times n$ array.

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


The array is not limited to $1$ or $2$ dimensions, but may conceptually have as many dimensions as may be needed.

It will, however, be noted that the visualizaton and depiction of an array of more than $2$ dimensions can offer challenges.


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 $\map a {i, j}$ 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 $\mathbf A = \sqbrk a_{m n}$ 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 $\sqbrk a_{n n}$ is usually denoted $\sqbrk a_n$.


Also known as

In the world of computer science, an array is often referred to as a subscripted variable.


Examples

Vector

A vector is an example of an array with $1$ dimension:

$\begin {pmatrix} a_1 \\ a_2 \\ \vdots \\ a_n \end {pmatrix}$


Matrix

A matrix is an example of an array with $2$ dimension:

$\begin {pmatrix} a_{1 1} & a_{1 2} & \cdots & a_{1 n} \\ a_{2 1} & a_{2 2} & \cdots & a_{2 n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m 1} & a_{m 2} & \cdots & a_{m n} \end {pmatrix}$


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

  • Results about arrays can be found here.


Sources

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