Matrices

An

m \times n

matrix: the

m

rows are horizontal and the

n

columns are vertical. Each element of a matrix is often denoted by a variable with two subscripts. For example, a_2,1 represents the element at the second row and first column of the matrix.

In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object.

For example,

{\begin{bmatrix}1&9&-13\\20&5&-6\end{bmatrix}}

is a matrix with two rows and three columns. This is often referred to as a "two by three matrix", a "

2\times3

-matrix", or a matrix of dimension

2\times3

.

Without further specifications, matrices represent linear maps, and allow explicit computations in linear algebra. Therefore, the study of matrices is a large part of linear algebra, and most properties and operations of abstract linear algebra can be expressed in terms of matrices. For example, matrix multiplication represents composition of linear maps.

Not all matrices are related to linear algebra. This is, in particular, the case in graph theory, of incidence matrices, and adjacency matrices. This article focuses on matrices related to linear algebra, and, unless otherwise specified, all matrices represent linear maps or may be viewed as such.

Square matrices, matrices with the same number of rows and columns, play a major role in matrix theory. Square matrices of a given dimension form a noncommutative ring, which is one of the most common examples of a noncommutative ring. The determinant of a square matrix is a number associated to 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 a polynomial determinant.

In geometry, matrices are widely used for specifying and representing 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 involves often to compute with matrices of huge dimension. Matrices are used in most areas of mathematics and most scientific fields, either directly, or through their use in geometry and numerical analysis.

Definition

A matrix is a rectangular array of numbers (or other mathematical objects), called the entries of the matrix. Matrices are subject to standard operations such as addition and multiplication. Most commonly, a matrix over a field F is a rectangular array of elements of F. 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 {A} ={\begin{bmatrix}-1.3&0.6\\20.4&5.5\\9.7&-6.2\end{bmatrix}}.

The numbers, symbols, or expressions in the matrix are called its entries or its elements. The horizontal and vertical lines of entries in a matrix are called rows and columns, respectively.

Size

The size of a matrix is defined by the number of rows and columns it contains. There is no limit to the numbers of rows and columns a matrix (in the usual sense) can have as long as they are positive integers. A matrix with m rows and n columns is called an m × n matrix, or m-by-n matrix, while m and n are called its dimensions. For example, the matrix A above is a 3 × 2 matrix.

Matrices with a single row are called row vectors, and those with a single column are called 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 vector	1 × n	${\begin{bmatrix}3&7&2\end{bmatrix}}$	A matrix with one row, sometimes used to represent a vector
Column vector	n × 1	${\begin{bmatrix}4\\1\\8\end{bmatrix}}$	A matrix with one column, sometimes used to represent a vector
Square matrix	n × 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

Matrices are commonly written in box brackets or parentheses:

\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}}={\begin{pmatrix}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{pmatrix}}=\left(a_{ij}\right)\in \mathbb {R} ^{m\times n}.

The specifics of symbolic matrix notation vary 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 (as in the case of ${\underline {\underline {A}}}$ ).

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[i,j] or A_i,j. For example, the (1,3) entry of the following matrix A is 5 (also denoted a₁₃, a_1,3, A[1,3] 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 the formula 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 parentheses. For example, the matrix above is defined as A = [i−j], 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 indicated using m × n as a subscript. For instance, the matrix A above is 3 × 4, and can be defined as A = [i − j] (i = 1, 2, 3; j = 1, ..., 4), or A = [i − j]_3×4.

Some programming languages utilize doubly subscripted arrays (or arrays of arrays) to represent an m-×-n matrix. Some programming languages start the numbering of array indexes 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.

An asterisk is occasionally used to refer to whole rows or columns in a matrix. For example, a_i,∗ refers to the i^th row of A, and a_∗,j refers to the j^th column of A.

The set of all m-by-n real matrices is often denoted ${\mathcal {M}}(m,n),$ or ${\mathcal {M}}_{m\times n}\mathbb {R} .$ The set of all m-by-n matrices matrices over another field or over a ring $R$ , is similarly denoted ${\mathcal {M}}(m,n,R),$ or ${\mathcal {M}}_{m\times n}(R).$ If m = n, that is, in the case of square matrices, one does not repeat the dimension: ${\mathcal {M}}(n,R),$ or ${\mathcal {M}}_{n}(R).$ Often, $M$ is used in place of ${\mathcal {M}}.$