Matrix (mathematics)

Notation

Matrices are commonly written in box brackets:

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

An alternative notation uses large parentheses instead of box brackets:

$\mathbf{A} = \left( \begin{array}{rrrr} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{array} \right).$

The specifics of symbolic matrix notation varies widely, with some prevailing trends. Matrices are usually symbolized using upper-case letters (such as A in the examples above), while the corresponding lower-case letters, with two subscript indices (e.g., a₁₁, or a_1,1), represent the entries. In addition to using upper-case letters to symbolize matrices, many authors use a special typographical style, commonly boldface upright (non-italic), to further distinguish matrices from other mathematical objects. An alternative notation involves the use of a double-underline with the variable name, with or without boldface style, (e.g., ).

The entry in the i-th row and j-th column of a matrix A is sometimes referred to as the i,j, (i,j), or (i,j)th entry of the matrix, and most commonly denoted as a_i,j, or a_ij. Alternative notations for that entry are A or A_i,j. For example, the (1,3) entry of the following matrix A is 5 (also denoted a₁₃, a_1,3, A or A_1,3):

$\mathbf{A}=\begin{bmatrix} 4 & -7 & \color{red}{5} & 0 \\ -2 & 0 & 11 & 8 \\ 19 & 1 & -3 & 12 \end{bmatrix}$

Sometimes, the entries of a matrix can be defined by a formula such as a_i,j = f(i, j). For example, each of the entries of the following matrix A is determined by a_ij = i − j.

$\mathbf A = \begin{bmatrix} 0 & -1 & -2 & -3\\ 1 & 0 & -1 & -2\\ 2 & 1 & 0 & -1 \end{bmatrix}$

In this case, the matrix itself is sometimes defined by that formula, within square brackets or double parenthesis. For example, the matrix above is defined as A =, or A = ((i-j)). If matrix size is m × n, the above-mentioned formula f(i, j) is valid for any i = 1, ..., m and any j = 1, ..., n. This can be either specified separately, or using m × n as a subscript. For instance, the matrix A above is 4 × 3 and can be defined as A = (i = 1, ..., 4; j = 1, 2, 3), or A = _4×3.

Some programming languages start the numbering of rows and columns at zero, in which case the entries of an m-by-n matrix are indexed by 0 ≤ i ≤ m − 1 and 0 ≤ j ≤ n − 1. This article follows the more common convention in mathematical writing where enumeration starts from 1.

The set of all m-by-n matrices is denoted (m, n).

Read more about this topic: Matrix (mathematics)

Matrix (mathematics) - Notation