Matrix (mathematics) explained

In mathematics, a matrix (: matrices) is a rectangular array of numbers or other mathematical objects with elements or entries arranged in rows and columns, usually satisfying certain properties of addition and multiplication.

For example, $\begin1 & 9 & -13 \\20 & 5 & -6 \end$ denotes a matrix with two rows and three columns. This is often referred to as a "two-by-three matrix", a matrix, or a matrix of dimension .

In linear algebra, matrices are used as linear maps. In geometry, matrices are used for geometric transformations (for example rotations) and coordinate changes. In numerical analysis, many computational problems are solved by reducing them to a matrix computation, and this often involves computing with matrices of huge dimensions. Matrices are used in most areas of mathematics and scientific fields, either directly, or through their use in geometry and numerical analysis.

Square matrices, matrices with the same number of rows and columns, play a major role in matrix theory. The determinant of a square matrix is a number associated with the matrix, which is fundamental for the study of a square matrix; for example, a square matrix is invertible if and only if it has a nonzero determinant and the eigenvalues of a square matrix are the roots of its characteristic polynomial,

\det(λI-A)

Matrix theory is the branch of mathematics that focuses on the study of matrices. It was initially a sub-branch of linear algebra, but soon grew to include subjects related to graph theory, algebra, combinatorics and statistics.

Definition

is a rectangular array of elements of . A real matrix and a complex matrix are matrices whose entries are respectively real numbers or complex numbers. More general types of entries are discussed below. For instance, this is a real matrix:

\mathbf = \begin-1.3 & 0.6 \\20.4 & 5.5 \\ 9.7 & -6.2\end.

The numbers (or other objects) in the matrix are called its entries or its elements. The horizontal and vertical lines of entries in a matrix are respectively called rows and columns.

Size

The size of a matrix is defined by the number of rows and columns it contains. There is no limit to the number of rows and columns that a matrix (in the usual sense) can have as long as they are positive integers. A matrix with rows and columns is called an matrix, or -by- matrix, where and are called its . For example, the matrix

} above is a matrix.

Matrices with a single row are called row matrices or row vectors, and those with a single column are called column matrices or column vectors. A matrix with the same number of rows and columns is called a square matrix. A matrix with an infinite number of rows or columns (or both) is called an infinite matrix. In some contexts, such as computer algebra programs, it is useful to consider a matrix with no rows or no columns, called an empty matrix.

Overview of a matrix size
Name	Size	Example	Description
Row matrix	1 x n	\begin{bmatrix}3&7&2\end{bmatrix}	A matrix with one row and more than one columns, sometimes used to represent a vector
Column matrix	n x 1	\begin{bmatrix}4\ 1\ 8\end{bmatrix}	A matrix with one column and more than one rows, sometimes used to represent a vector
	n x n	\begin{bmatrix} 9&13&5\\ 1&11&7\\ 2&6&3 \end{bmatrix}	A matrix with the same number of rows and columns, sometimes used to represent a linear transformation from a vector space to itself, such as reflection, rotation, or shearing.

Notation

The specifics of symbolic matrix notation vary widely, with some prevailing trends. Matrices are commonly written in square brackets or parentheses, so that an matrix

is represented as

\mathbf =\begina_ & a_ & \cdots & a_ \\a_ & a_ & \cdots & a_ \\\vdots & \vdots & \ddots & \vdots \\a_ & a_ & \cdots & a_\end =\begina_ & a_ & \cdots & a_ \\a_ & a_ & \cdots & a_ \\\vdots & \vdots & \ddots & \vdots \\a_ & a_ & \cdots & a_\end.

This may be abbreviated by writing only a single generic term, possibly along with indices, as in

\mathbf = \left(a_\right), \quad \left[a_{ij}\right], \quad \text \quad \left(a_\right)_

A=(a_i,j)_1\leq

in the case that .

Matrices are usually symbolized using upper-case letters (such as

} in the examples above), while the corresponding lower-case letters, with two subscript indices (e.g.,, or), represent the entries. In addition to using upper-case letters to symbolize matrices, many authors use a special typographical style, commonly boldface roman (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, as in .

The entry in the th row and th column of a matrix is sometimes referred to as the

{i,j}

(i,j)

entry of the matrix, and commonly denoted by

a_i,j

or . Alternative notations for that entry are

{A[i,j]}

and . For example, the

(1,3)

entry of the following matrix

is (also denoted,,

A[1,3]

or):

\mathbf=\begin 4 & -7 & \color & 0 \\-2 & 0 & 11 & 8 \\19 & 1 & -3 & 12\end

Sometimes, the entries of a matrix can be defined by a formula such as . For example, each of the entries of the following matrix

is determined by the formula .

\mathbf A = \begin0 & -1 & -2 & -3\\1 & 0 & -1 & -2\\2 & 1 & 0 & -1\end

In this case, the matrix itself is sometimes defined by that formula, within square brackets or double parentheses. For example, the matrix above is defined as

}=[i-j] or . If matrix size is, the above-mentioned formula

f(i,j)

is valid for any

i=1,...,m

and any . This can be specified separately or indicated using as a subscript. For instance, the matrix

above is, and can be defined as

}=[i-j] (i=1,2,3; j=1,\dots,4) or .

Some programming languages utilize doubly subscripted arrays (or arrays of arrays) to represent an