Definition:Array

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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

  • Results about arrays can be found here.


Sources

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