Matrix (mathematics)

In mathematics, a matrix (pl.: 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, \[\begin{bmatrix}1 & 9 & -13 \\20 & 5 & -6 \end{bmatrix}\] denotes a matrix with two rows and three columns. This is often referred to as a "two-by-three matrix", a 2 × 3 matrix, or a matrix of dimension 2 × 3.

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. Also, the eigenvalues of a square matrix are the roots of its characteristic polynomial, \(\det(\lambda I - A)\).

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.

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 (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.

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 m rows and n columns is called an m × n matrix, or m-by-n matrix, where m and n are called its dimensions. For example, the matrix \({\mathbf{A}}\) above is a 3 × 2 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.

The specifics of symbolic matrix notation vary widely, with some prevailing trends. Matrices are commonly written in square brackets or parentheses, so that an m × n matrix \(\mathbf{A}\) is represented as \[\mathbf{A} = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix} = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{pmatrix}.\] This may be abbreviated by writing only a single generic term, possibly along with indices, as in \[\mathbf{A} = \left(a_{ij}\right), \quad \left[ a_{ij}\right], \quad \text{or} \quad \left(a_{ij}\right)_{1\leq i\leq m, \; 1\leq j\leq n}\] or \(\mathbf{A}=(a_{i,j})_{1\leq i,j\leq n}\) in the case that ⁠\(n=m\)⁠.

Matrices are usually symbolized using upper-case letters (such as \({\mathbf{A}}\) in the examples above), while the corresponding lower-case letters, with two subscript indices (e.g., ⁠\(a_{11}\)⁠, 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 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 ⁠\(\underline{\underline{A} }\)⁠.

The entry in the ith row and jth column of a matrix A is sometimes referred to as the \({i,j}\) or \((i,j)\) entry of the matrix, and commonly denoted by \(a_{i,j}\) or ⁠\(a_{ij}\)⁠. Alternative notations for that entry are \({\mathbf{A}[i,j]}\) and ⁠\(\mathbf{A}_{i,j}\)⁠. For example, the \((1,3)\) entry of the following matrix \(\mathbf{A}\) is 5 (also denoted ⁠\(a_{1 3}\)⁠, ⁠\(a_{1,3}\)⁠, \(\mathbf{A}[1,3]\) or ⁠\({\mathbf{A} }_{1,3}\)⁠): \[\mathbf{A}=\begin{bmatrix} 4 & -7 & \color{red}{5} & 0 \\ -2 & 0 & 11 & 8 \\ 19 & 1 & -3 & 12 \end{bmatrix}\]

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 \(\mathbf{A}\) 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 \({\mathbf{A}}=[i-j]\) or ⁠\(\mathbf{A}=((i-j))\)⁠. If matrix size is m × n, the above-mentioned formula \(f(i,j)\) is valid for any \(i=1,\dots,m\) and any ⁠\(j=1,\dots,n\)⁠. This can be specified separately or indicated using m × n as a subscript. For instance, the matrix \(\mathbf{A}\) above is 3 × 4, and can be defined as \({\mathbf{A}}=[i-j] (i=1,2,3; j=1,\dots,4)\) or ⁠\(\mathbf{A}=[i-j]_{3\times4}\)⁠.

Condensed: the full section is in Wikipedia.

Several basic operations can be applied to matrices. Some, such as transposition and submatrix do not depend on the nature of the entries. Others, such as matrix addition, scalar multiplication, matrix multiplication, and row operations involve operations on matrix entries and therefore require that matrix entries are numbers or belong to a field or a ring.

In this section, it is supposed that matrix entries belong to a fixed ring, which is typically a field of numbers.

Matrix addition and subtraction require matrices of a consistent size, and are calculated entrywise. The sum A + B and the difference A − B of two m × n matrices are:

\[\begin{aligned} (\bold A + \bold B)_{i, j} = \bold A_{i,j} + \bold B_{i,j}, \quad 1 \leq i \leq m, \quad 1 \leq j \leq n. \\ (\bold A - \bold B)_{i, j} = \bold A_{i,j} - \bold B_{i,j}, \quad 1 \leq i \leq m, \quad 1 \leq j \leq n. \end{aligned}\]

For example,

\[\begin{bmatrix} 1 & 3 & 1 \\ 1 & 0 & 0 \end{bmatrix} + \begin{bmatrix} 0 & 0 & 5 \\ 7 & 5 & 0 \end{bmatrix} = \begin{bmatrix} 1+0 & 3+0 & 1+5 \\ 1+7 & 0+5 & 0+0 \end{bmatrix} = \begin{bmatrix} 1 & 3 & 6 \\ 8 & 5 & 0 \end{bmatrix}\]

Familiar properties of numbers extend to these operations on matrices: for example, addition is commutative, that is, the matrix sum does not depend on the order of the summands: A + B = B + A.

The product cA of a number c (also called a scalar in this context) and a matrix A is computed by multiplying each entry of A by c: \[(c\bold A)_{i,j} = c \cdot \bold A_{i,j}\] This operation is called scalar multiplication, but its result is not named "scalar product" to avoid confusion, since "scalar product" is often used as a synonym for "inner product". For example:

\[2 \cdot \begin{bmatrix} 1 & 8 & -3 \\ 4 & -2 & 5 \end{bmatrix} = \begin{bmatrix} 2 \cdot 1 & 2\cdot 8 & 2\cdot -3 \\ 2\cdot 4 & 2\cdot -2 & 2\cdot 5 \end{bmatrix} = \begin{bmatrix} 2 & 16 & -6 \\ 8 & -4 & 10 \end{bmatrix}\]

Matrix subtraction is consistent with composition of matrix addition with scalar multiplication by –1:

\[\mathbf A - \mathbf B = \mathbf A +(-1)\cdot \mathbf B\]

The transpose of an m × n matrix A is the n × m matrix A (also denoted A or A) formed by turning rows into columns and vice versa: \[\left( \bold A^{\rm T} \right)_{i,j} = \bold A_{j,i}.\] For example: \[\begin{bmatrix} 1 & 2 & 3 \\ 0 & -6 & 7 \end{bmatrix}^\mathrm{T} = \begin{bmatrix} 1 & 0 \\ 2 & -6 \\ 3 & 7 \end{bmatrix}\]

The transpose is compatible with addition and scalar multiplication, as expressed by (cA) = c(A) and (A + B) = A + B. Finally, (A) = A.

Multiplication of two matrices corresponds to the composition of linear transformations represented by each matrix. It is defined if and only if the number of columns of the left matrix is the same as the number of rows of the right matrix. If A is an m × n matrix and B is an n × p matrix, then their matrix product AB is the m × p matrix whose entries are given by the dot product of the corresponding row of A and the corresponding column of B: \[[\mathbf{AB}]_{i,j} = a_{i,1}b_{1,j} + a_{i,2}b_{2,j} + \cdots + a_{i,n}b_{n,j} = \sum_{r=1}^n a_{i,r}b_{r,j},\] where 1 ≤ i ≤ m and 1 ≤ j ≤ p. For example, the underlined entry 2340 in the product is calculated as (2 × 1000) + (3 × 100) + (4 × 10) = 2340: \[\begin{aligned} \begin{bmatrix} \underline{2} & \underline 3 & \underline 4 \\ 1 & 0 & 0 \\ \end{bmatrix} \begin{bmatrix} 0 & \underline{1000} \\ 1 & \underline{100} \\ 0 & \underline{10} \\ \end{bmatrix} &= \begin{bmatrix} 3 & \underline{2340} \\ 0 & 1000 \\ \end{bmatrix}. \end{aligned}\]

Matrix multiplication satisfies the rules (AB)C = A(BC) (associativity), and (A + B)C = AC + BC as well as C(A + B) = CA + CB (left and right distributivity), whenever the size of the matrices is such that the various products are defined. The product AB may be defined without BA being defined, namely if A and B are m × n and n × k matrices, respectively, and m ≠ k. Even if both products are defined, they generally need not be equal, that is: \[\bold{AB} \ne \bold{BA}.\]

In other words, matrix multiplication is not commutative, in marked contrast to (rational, real, or complex) numbers, whose product is independent of the order of the factors. An example of two matrices not commuting with each other is: \[\begin{bmatrix} 1 & 2\\ 3 & 4\\ \end{bmatrix} \begin{bmatrix} 0 & 1\\ 0 & 0\\ \end{bmatrix}= \begin{bmatrix} 0 & 1\\ 0 & 3\\ \end{bmatrix},\] whereas \[\begin{bmatrix} 0 & 1\\ 0 & 0\\ \end{bmatrix} \begin{bmatrix} 1 & 2\\ 3 & 4\\ \end{bmatrix}= \begin{bmatrix} 3 & 4\\ 0 & 0\\ \end{bmatrix}.\]

Besides the ordinary matrix multiplication just described, other less frequently used operations on matrices that can be considered forms of multiplication also exist, such as the Hadamard product and the Kronecker product. They arise in solving matrix equations such as the Sylvester equation.

There are three types of row operations:

1. row addition, that is, adding a row to another.
2. row multiplication, that is, multiplying all entries of a row by a non-zero constant;
3. row switching, that is, interchanging two rows of a matrix;

These operations are used in several ways, including solving linear equations and finding matrix inverses with Gauss elimination and Gauss-Jordan elimination, respectively.

A submatrix of a matrix is a matrix obtained by deleting any collection of rows or columns or both. For example, from the following 3 × 4 matrix, we can construct a 2 × 3 submatrix by removing row 3 and column 2: \[\mathbf{A}=\begin{bmatrix} 1 & \color{red}{2} & 3 & 4 \\ 5 & \color{red}{6} & 7 & 8 \\ \color{red}{9} & \color{red}{10} & \color{red}{11} & \color{red}{12} \end{bmatrix} \rightarrow \begin{bmatrix} 1 & 3 & 4 \\ 5 & 7 & 8 \end{bmatrix}.\]

The minors and cofactors of a matrix are found by computing the determinant of certain submatrices.

A principal submatrix is a square submatrix obtained by removing certain rows and columns. The definition varies from author to author. According to some authors, a principal submatrix is a submatrix in which the set of row indices that remain is the same as the set of column indices that remain. Other authors define a principal submatrix as one in which the first k rows and columns, for some number k, are the ones that remain; this type of submatrix has also been called a leading principal submatrix.

Matrices can be used to compactly write and work with multiple linear equations, that is, systems of linear equations. For example, if A is an m × n matrix, x designates a column vector (that is, n × 1 matrix) of n variables x₁, x₂, ..., x_n, and b is an m × 1 column vector, then the matrix equation \[\mathbf{Ax} = \mathbf{b}\] is equivalent to the system of linear equations \[\begin{aligned} a_{1,1}x_1 + a_{1,2}x_2 + &\cdots + a_{1,n}x_n = b_1 \\ &\ \ \vdots \\ a_{m,1}x_1 + a_{m,2}x_2 + &\cdots + a_{m,n}x_n = b_m \end{aligned}\]

Using matrices, this can be solved more compactly than would be possible by writing out all the equations separately. If n = m and the equations are independent, then this can be done by writing \[\mathbf{x} = \mathbf{A}^{-1} \mathbf{b}\] where A is the inverse matrix of A. If A has no inverse, solutions, if any, can be found using its generalized inverse.

Matrices and matrix multiplication reveal their essential features when related to linear transformations, also known as linear maps. A real m-by-n matrix A gives rise to a linear transformation \(\R^n \to \R^m\) mapping each vector x in ⁠\(\R^n\)⁠ to the (matrix) product Ax, which is a vector in ⁠\(\R^m.\)⁠ Conversely, each linear transformation \(f: \R^n \to \R^m\) arises from a unique m-by-n matrix A: explicitly, the (i, j)-entry of A is the ith coordinate of f (e_j), where e_j = (0, ..., 0, 1, 0, ..., 0) is the unit vector with 1 in the jth position and 0 elsewhere. The matrix A is said to represent the linear map f, and A is called the transformation matrix of f.

For example, the 2 × 2 matrix \[\mathbf{A} = \begin{bmatrix} a & c\\b & d \end{bmatrix}\] can be viewed as the transform of the unit square into a parallelogram with vertices at (0, 0), (a, b), (a + c, b + d), and (c, d). The parallelogram pictured at the right is obtained by multiplying A with each of the column vectors ⁠\(\left[\begin{smallmatrix} 0 \\ 0 \end{smallmatrix}\right]\)⁠, ⁠\(\left[\begin{smallmatrix} 1 \\ 0 \end{smallmatrix}\right]\)⁠, ⁠\(\left[\begin{smallmatrix} 1 \\ 1 \end{smallmatrix}\right]\)⁠, and ⁠\(\left[\begin{smallmatrix} 0 \\ 1 \end{smallmatrix}\right]\)⁠ in turn. These vectors define the vertices of the unit square. The following table shows several 2 × 2 real matrices with the associated linear maps of ⁠\(\R^2\)⁠. The blue original is mapped to the green grid and shapes. The origin (0, 0) is marked with a black point.

Under the 1-to-1 correspondence between matrices and linear maps, matrix multiplication corresponds to composition of maps: if a k-by-m matrix B represents another linear map ⁠\(g: \R^m \to \R^k\)⁠, then the composition g ∘ f is represented by BA since \[(g \circ f)(\bold x) = g(f(\bold x)) = g(\bold{Ax}) = \bold B(\bold {Ax}) = (\bold{BA}) \bold x.\]

The last equality follows from the above-mentioned associativity of matrix multiplication.

The rank of a matrix A is the maximum number of linearly independent row vectors of the matrix, which is the same as the maximum number of linearly independent column vectors. Equivalently it is the dimension of the image of the linear map represented by A. The rank, nullity theorem states that the dimension of the kernel of a matrix plus the rank equals the number of columns of the matrix.

A square matrix is a matrix with the same number of rows and columns. An n-by-n matrix is known as a square matrix of order n. Any two square matrices of the same order can be added and multiplied. The entries a_ii form the main diagonal of a square matrix. They lie on the imaginary line running from the top left corner to the bottom right corner of the matrix.

Square matrices of a given dimension form a noncommutative ring, which is one of the most common examples of a noncommutative ring.