Matrix (mathematics)

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In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. Matrices are used to describe linear equations, keep track of the coefficients of linear transformations and to record data that depend on multiple parameters. Matrices are described by the field of matrix theory. They can be added, multiplied, and decomposed in various ways, which also makes them a key concept in the field of linear algebra.

Organization of a matrix

Definition

A matrix is a rectangular arrangement of numbers.^[1] For example,

$\mathbf{X} = \begin{bmatrix} 9 & 8 & 6 \\ 1 & 2 & 7 \\ 4 & 9 & 2 \\ 6 & 0 & 5 \end{bmatrix}$ alternatively denoted using parentheses instead of box brackets: $\begin{pmatrix} 9 & 8 & 6 \\ 1 & 2 & 7 \\ 4 & 9 & 2 \\ 6 & 0 & 5 \end{pmatrix}$

The horizontal lines in a matrix are called rows and the vertical lines are called columns. The numbers in the matrix are called its entries. To specify a matrix' size, a matrix with m rows and n columns is called an m-by-n matrix or m × n matrix. m and n are called its dimensions. The above is a 4-by-3 matrix.

A matrix where one of the dimensions equals one is often called a vector, and may be interpreted as an element of real coordinate space. An m × 1 matrix (one column and m rows) is called a column vector and a 1 × n matrix (one row and n columns) is called a row vector. For example, the second row vector of the above matrix is

$\begin{bmatrix} 1 & 2 & 7 \\ \end{bmatrix}$

Most of this article focusses on real and complex matrices, i.e., matrices whose entries are real or complex numbers. More general types of entries are discussed below.

Notation

The entry that lies in the i-th row and the j-th column of a matrix is typically referred to as the i,j, (i,j), or (i,j)^th entry of the matrix. For example, (2,3) entry of the above matrix X is 7.

Matrices are usually denoted using upper-case letters, while the corresponding lower-case letters, with two subscript indices, represent the entries. For example, the (i, j)^th entry of a matrix A is most commonly written as a_i,j. Alternative notations for that entry are A[i,j] or A_i,j. 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 variables. Following this convention, A is a matrix, distinguished from A, a scalar. An alternate convention is to annotate matrices with their dimensions in small type underneath the symbol, for example, $\underset{m \times n}{A}$ for an m-by-n matrix. The set of all m-by-n matrices is often denoted M(m, n).

A common shorthand is

A = (a_i,j)_{i=1,...,m; j=1,...,n} or more briefly A = (a_i,j)_m×n

to define an m × n matrix A. In this case, the entries a_i,j are defined separately for all integers 1 ≤ i ≤ m and 1 ≤ j ≤ n; for example the 2-by-2 matrix

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

is be specified by A = (i − j)_{i=1,2; j=1,2}

In some programming languages the numbering of rows and columns starts 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, but this article will follow the numerotation starting from 1.

History

The study of matrices is quite old. A 3-by-3 magic square appears in the Chinese literature dating from as early as 650 BC.^[2]

Matrices have a long history of application in solving linear equations. An important Chinese text from between 300 BC and AD 200, The Nine Chapters on the Mathematical Art (Jiu Zhang Suan Shu), is the first example of the use of matrix methods to solve simultaneous equations.^[3] In the seventh chapter, "Too much and not enough," the concept of a determinant first appears almost 2000 years before its publication by the Japanese mathematician Seki Kowa in 1683 and the German mathematician Gottfried Leibniz in 1693.

Magic squares were known to Arab mathematicians, possibly as early as the 7th century, when the Arabs conquered northwestern parts of the Indian subcontinent and learned Indian mathematics and astronomy, including other aspects of combinatorial mathematics. It has also been suggested that the idea came via China. The first magic squares of order 5 and 6 appear in an encyclopedia from Baghdad circa 983 AD, the Encyclopedia of the Brethren of Purity (Rasa'il Ihkwan al-Safa); simpler magic squares were known to several earlier Arab mathematicians.^[2]

The modern matrix concept started with linear algebra. Later, after the development of the theory of determinants by Seki Kowa and Leibniz in the late 17th century, Cramer developed the theory further in the 18th century, presenting Cramer's rule in 1750. Carl Friedrich Gauss and Wilhelm Jordan developed Gauss-Jordan elimination in the 1800s.

The term "matrix" was coined in 1848 by J. J. Sylvester. Cayley, Hamilton, Grassmann, Frobenius and von Neumann are among the famous mathematicians who have worked on matrix theory.

Basic operations

Main articles: Matrix addition, Scalar multiplication, and Transpose

There are a number of operations that can be applied to modify matrices called matrix addition, scalar multiplication and transposition. These form the basic techniques to deal with matrices.

Operation	Definition	Example
Addition	Given m-by-n matrices A and B, their sum A+B is calculated entrywise, i.e. (A + B)_i,j = A_i,j + B_i,j, where 1 ≤ i ≤ m and 1 ≤ j ≤ n.^[4]	$\begin{bmatrix} 1 & 3 & 1 \\ 1 & 0 & 0 \end{bmatrix} + \begin{bmatrix} 0 & 0 & 5 \\ 7 & 5 & 0 \end{bmatrix} = \begin{bmatrix} 1+0 & 3+0 & 1+5 \\ 1+7 & 0+5 & 0+0 \end{bmatrix} = \begin{bmatrix} 1 & 3 & 6 \\ 8 & 5 & 0 \end{bmatrix}.$
Scalar multiplication	Given a matrix A and a number (also called a scalar in the parlance of abstract algebra) c, the scalar multiplication cA is given by multiplying every entry of A by c: (cA)_i,j = c · A_i,j.^[5]	$2 \cdot \begin{bmatrix} 1 & 8 & -3 \\ 4 & -2 & 5 \end{bmatrix} = \begin{bmatrix} 2 \cdot 1 & 2\cdot 8 & 2\cdot -3 \\ 2\cdot 4 & 2\cdot -2 & 2\cdot 5 \end{bmatrix} = \begin{bmatrix} 2 & 16 & -6 \\ 8 & -4 & 10 \end{bmatrix}.$
Transpose	The transpose of an m-by-n matrix A is the n-by-m matrix A^T (also denoted by A^tr or ^tA) formed by turning rows into columns and columns into rows: (A^T)_i,j = A_j,i.^[6]	$\begin{bmatrix} 1 & 3 & 1 \\ 0 & -6 & 0 \end{bmatrix}^T = \begin{bmatrix} 1 & 0 \\ 3 & -6 \\ 1 & 0 \end{bmatrix}$

Familiar properties of numbers extend to these operations of matrices: for example, addition is commutative, i.e. the matrix sum does not depend on the order of the summands: A + B = B + A.^[7] The transpose, which does not exist for numbers, is also well compatible with the addition and scalar multiplication, as expressed by (cA)^T = c(A^T), as well as (A + B)^T = A^T + B^T. Finally, (A^T)^T = A.

Matrix multiplication, linear equations and linear transformations

Main article: Matrix multiplication

Schematic depiction of the matrix product AB of two matrices A and B.

Multiplication of two matrices is well-defined 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-by-n matrix and B is an n-by-p matrix, then their matrix product AB is the m-by-p matrix whose entries are given by

$(\mathbf{AB})_{i,j} = \sum_{r=1}^n A_{i,r}B_{r,j}$

where 1 ≤ i ≤ m and 1 ≤ j ≤ p.^[8] For example (the highlighted entry 1 in the product is calculated as the product 1 · 1 + 0 · 1 + 2 · 0 = 1):

$\begin{align} \begin{bmatrix} \underline{1} & \underline 0 & \underline 2 \\ -1 & 3 & 1 \\ \end{bmatrix} \times \begin{bmatrix} 3 & \underline 1 \\ 2 & \underline 1 \\ 1 & \underline 0 \\ \end{bmatrix} &= \begin{bmatrix} 5 & \underline 1 \\ 4 & 2 \\ \end{bmatrix}. \end{align}$

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.^[9] The product AB may be defined without BA being defined, namely if A and B are m-by-n and n-by-k matrices, respectively, and m ≠ k. Even if both products are defined, they need not be equal, i.e. generally one has

AB ≠ BA,

i.e. 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 for two matrices not commuting with each other is: $\begin{bmatrix} 1 & 2\\ 3 & 4\\ \end{bmatrix}\cdot \begin{bmatrix} 0 & 1\\ 0 & 0\\ \end{bmatrix}= \begin{bmatrix} 0 & 1\\ 0 & 3\\ \end{bmatrix}$ , whereas $\begin{bmatrix} 0 & 1\\ 0 & 0\\ \end{bmatrix}\cdot \begin{bmatrix} 1 & 2\\ 3 & 4\\ \end{bmatrix}= \begin{bmatrix} 3 & 4\\ 0 & 0\\ \end{bmatrix}.$

The identity matrix I_n of size n is the n-by-n matrix in which all the elements on the main diagonal are equal to 1 and all other elements are equal to 0, e.g.

$\mathbf{I}_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} .$

It is called identity matrix because multiplication with it leaves a matrix unchanged: MI_n = M and I_mM = M for any m-by-n matrix M.

Besides the ordinary matrix multiplication just described, there exist other less frequently used operations on matrices that can be considered forms of multiplication, such as the Hadamard product and the Kronecker product.

Linear equations

Main articles: Linear equation and System of linear equations

A particular case of matrix multiplication is tightly linked to linear equations: if x designates a column vector (i.e. n×1-matrix) of n variables x₁, x₂, ..., x_n, and A is an m-by-n matrix, then the matrix equation

Ax = b, where b is some m×1-column vector

is equivalent to the system of linear equations

A_1,1x₁ + A_1,2x₂ + ... + A_1,nx_n = b₁

...

A_m,1x₁ + A_m,2x₂ + ... + A_m,nx_n = b_m^[10]

This way, matrices can be used to compactly write and deal with multiple linear equations, i.e. systems of linear equations.

Linear transformations

Main articles: Linear transformation and Transformation matrix

Matrices and matrix multiplication reveal their essential features when related to linear transformations (or linear maps). Any n-by-m matrix A gives rise to a linear transformation Rⁿ → R^m, by assigning to any vector x in Rⁿ the (matrix) product Ax, which is an element in R^m. Conversely, given any linear transformation, there exists a unique matrix A, such that the transformation is given by this formula: its (i, j)-entry is given by the j-th component of f(e_i), where e_i = (0, ..., 0, 1, 0, ..., 0) is the unit vector with 1 at the i-th spot and 0 elsewhere. The matrix A is said to represent the linear map f, or that A is the transformation matrix of f.

The following table shows a number of 2-by-2 matrices along with the associated linear maps of R². The blue original is mapped to the green grid and shapes, the origin (0, 0) is marked with a black point.

Vertical shear with m=1.25. $\begin{bmatrix} 1 & 1.25 \\ 0 & 1 \end{bmatrix}$	Horizontal flip $\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}$	Squeeze mapping with r=1.5 $\begin{bmatrix} 3/2 & 0 \\ 0 & 2/3 \end{bmatrix}$	Scaling by a factor of 1.5 $\begin{bmatrix} 3/2 & 0 \\ 0 & 3/2 \end{bmatrix}$	Rotation by π/6 = 30° $\begin{bmatrix}\cos(\pi / 6) & -\sin(\pi / 6)\\ \sin(\pi / 6) & \cos(\pi / 6)\end{bmatrix}$

Matrix multiplication corresponds to the composition of maps. If the k-by-m matrix B represents another linear map g : R^m → R^k, then the composition g ∘ f is represented by BA:

(g ∘ f)(x) = g(f(x)) = g(Ax) = B(Ax) = (BA)x,

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

The nullity and rank of a matrix A are the dimension of the kernel and image of the linear map represented by A, respectively. The rank is also characterized as the number of linear independent row vectors of the matrix, which is the same as the number of linear independent column vectors.^[11] The sum of the nullity and rank equals the number of columns of the matrix, by the rank-nullity theorem.^[12]

Square matrices

A square matrix is a matrix which has the same number of rows and columns. Due to this size restriction, all matrices can be multiplicated (and added). A n-by-n matrix A is called invertible or non-singular if there exists a matrix B such that

AB = I_n.^[13]

This implies that BA = I_n, too.^[14] Moreover, B is unique and is called the inverse matrix of A, denoted by A⁻¹.

The entries A_i,i form the main diagonal of a matrix. If all entries outside the main diagonal are zero, A is called diagonal matrix. If only all entries above (below) the main diagonal are zero, A is called a lower triangular matrix (upper triangular matrix, respectively). For example, if n = 3, they look like

$\begin{bmatrix} d_{11} & 0 & 0 \\ 0 & d_{22} & 0 \\ 0 & 0 & d_{33} \\ \end{bmatrix}$ (diagonal), $\begin{bmatrix} l_{11} & 0 & 0 \\ l_{21} & l_{22} & 0 \\ l_{31} & l_{32} & l_{33} \\ \end{bmatrix}$ (lower) and $\begin{bmatrix} u_{11} & u_{12} & u_{13} \\ 0 & u_{22} & u_{23} \\ 0 & 0 & u_{33} \\ \end{bmatrix}$ (upper triangular matrix).

Determinant

Main article: Determinant

A linear transformation on R² given by a matrix. The determinant of this matrix is −1, as the area of the green parallelogram at the right is 1, but the map reverses the orientation, since it turns the counterclockwise orientation of the vectors to a clockwise one.

The determinant det(A) of a square matrix A is a number encoding certain properties of the matrix. The matrix is invertible if and only if the determinant is nonzero. Its absolute value equals the area (in R²) or volume (in R³) of the image of the unit square (or cube), while its sign corresponds to the orientation of the corresponding linear map: the determinant is positive if and only if the orientation is preserved.

The determinant of 2-by-2 matrices $\begin{pmatrix}a&b\\c&d\end{pmatrix}$ is given by ad − bc. For 3-by-3 matrices, the expression involves 6 terms (rule of Sarrus); alternatively, the Laplace expansion can be used. These two cases are generalised by the more lengthy Leibniz formula to all dimensions.^[15] Alternatively, determinants can be calculated by transforming the matrix into an upper or lower triangular matrix, by adding suitable multiples of rows or columns to other rows, which does not change the determinant, and by exchanging rows, which alters the sign of the determinant.^[16] The determinant of a lower or upper triangular matrix (in particular of diagonal matrices) equals the product of the entries on the main diagonal. Finally, the Laplace expansion expresses the determinant in terms of determinants of smaller matrices.

The determinant of a product of matrices equals the product of the determinants: det(AB) = det(A) · det(B).^[17] Determinants can be used to solve linear systems using Cramer's rule, which expresses the solutions to linear systems in an explicit way in terms of certain determinants related to the system.^[18]

Eigenvalues and eigenvectors

Main article: Eigenvalues and eigenvectors

A number λ and a non-zero vector v satisfying

Av = λv,

are called eigenvalue and eigenvector of A, respectively.^{[nb 1]}^[19] The number λ is an eigenvalue of A if and only if A−λI_n is not invertible, which happens if and only if

det(A−λI) = 0.^[20]

The function p_A(t) = det(A−tI) is called the characteristic polynomial of A, the degree of this polynomial is n. Therefore p_A(t) has at most n (possibly complex) different roots, i.e. eigenvalues of the matrix.^[21]

The trace of a square matrix is the sum of its diagonal entries. It equals the sum of its n eigenvalues.

Definiteness

A square matrix A that is is equal to its transpose, i.e. A = A^T, is a symmetric matrix; if it is equal to the negative of its transpose, i.e. A = −A^T, then it is a skew-symmetric matrix. A symmetric n×n-matrix is called positive definite (negative definite, resp.), if for all nonzero vectors x ∈ Rⁿ the associated quadratic form given by

Q(x) = x^TAx

takes only positive values (negative, respectively). An analogous notion exists for complex Hermitian matrices, which satisfy A^∗ = A, where the star denotes the conjugate transpose of the matrix, i.e. the transpose of the complex conjugate of A. The following table shows two possibilities for 2-by-2 matrices:

Matrix A	Quadratic form Q_A(v), v = (x, y)	Definiteness	Set of vectors v, such that Q_A(v) = 1.
$\begin{bmatrix} 1/4 & 0\\ 0 & 1\end{bmatrix}$	1/4 x² + y²	positive definite	Ellipse
$\begin{bmatrix} 1/4 & 0\\ 0 & -1/4\end{bmatrix}$	1/4 x² − 1/4 y²	indefinite	Hyperbola

By the spectral theorem, positive definite matrices have an eigenbasis, i.e. every vector is expressible as a linear combination of eigenvectors. Also, all eigenvalues are real. The function

(x, y) ↦ x^TAy

is the bilinear form associated to A.

Computational aspects

In addition to theoretical knowledge of properties of matrices and their relation to other fields, it is important for practical purposes to perform matrix calculations effectively and precisely. The domain studying these matters is called numerical linear algebra.^[22]

An important point is the performance of algorithms, i.e. upper bounds of how many elementary operations such as additions and multiplications of scalars are necessary to perform some operation matrices. For example, calculating the matrix product of two n-by-n matrix using the definition given above needs n³ multiplications, since for any of the n² entries of the product, n multiplications are necessary. The Strassen algorithm outperforms this "naive" algorithm; it needs only n^2.807 multiplications.^[23] A refined approach also incorporates specific features of the computing devices.

Another issue is numerical stability which means avoiding for example very small numbers, where rounding effects inherent to many computer calculations are particularly critical. For example, calculating the inverse of a matrix via Laplace's formula (Adj (A) denotes the adjugate matrix of A)

A⁻¹ = Adj(A) / det(A)

may lead to significant rounding errors if the determinant of the matrix is very small. The norm of a matrix can be used to capture the conditioning of linear algebraic problems attached to the matrix, such as computing the inverse.^[24]

In many practical situations, additional information about the matrices involved is known. An important case are sparse matrices, i.e. matrices most of whose entries are zero. There are specifically adapted algorithms for, say, solving linear systems Ax = b for sparse matrices A, such as the conjugate gradient method.^[25] It can be implemented both directly or iteratively, i.e. finding a sequence of vectors x_n converging to a solution of the equation.

Although most computer languages are not designed with commands or libraries for matrices, as early as the 1970s, some engineering desktop computers such as the HP 9830 had ROM cartridges to add BASIC commands for matrices. Some computer languages such as APL were designed to manipulate matrices, and various mathematical programs can be used to aid computing with matrices.

Matrix decomposition methods

Main article: Matrix decomposition

There are several methods to render matrices into a more easily accessible form. They are generally referred to as matrix transformation or matrix decomposition techniques. The interest of all these decomposition techniques is that they preserve certain properties of the matrices in question, such as determinant, ranks or inverses, so that these quantities can be calculated after applying the transformation, or that certain matrix operations are algorithmically easier to carry out for some types of matrices.

The LU decomposition factors matrices as a product of lower (L) and an upper triangular matrices (U). Once this decomposition is calculated, linear systems can be solved more efficiently, by a simple technique called forward and back substitution. Likewise, inverses of triangular matrices are algorithmically easier to calculate. The Gaussian elimination is a similar algorithm; it transforms any matrix to row echelon form. Both methods proceed by multiplying the matrix by suitable elementary matrices, which correspond to permuting rows or columns and adding multiples of one row to another row.

The eigendecomposition expresses A as a product VDV⁻¹, where D is a diagonal matrix and V is a suitable invertible matrix. If A can be written in this form, it is called diagonalizable. More generally, and applicable to all matrices, the Jordan decomposition transforms a matrix into Jordan normal form. To avoid numerically ill-conditioned situations, further algorithms such as the Schur decomposition can be employed. For example, given the eigendecomposition, the n^th power of A (i.e. n-fold iterated matrix multiplication) can be calculated via

Aⁿ = (VDV⁻¹)ⁿ = VDV⁻¹VDV⁻¹...VDV⁻¹ = VDⁿV⁻¹

and the power of a diagonal matrix can be calculated by taking the corresponding powers of the diagonal entries, which is much easier than doing the exponentiation for A instead. This can be used to compute the matrix exponential e^A, a need frequently arising in solving linear differential equations, matrix logarithms and square roots of matrices.

The QR decomposition expresses a matrix as the product of an orthogonal matrix and a upper triangular matrix. It can be used for the QR algorithm, one of several algorithms computing the eigenvalues of a matrix.

Abstract algebraic aspects

Matrices with more general entries

This article focuses on matrices whose entries are real or complex numbers. However, matrices can be considered with much more general types of entries than real or complex numbers. As a first step of generalization, any field F, i.e. a set where addition, subtraction, multiplication and division operations are defined and well-behaved, may be used instead of R or C, for example rational numbers or finite fields. Wherever eigenvalues are considered, the choice of the field usually matters insofar as a the characteristic polynomial, despite having real coefficients may have complex solutions. Therefore, the field is often required to be C or any algebraically closed field when such issues arise.

n-by-m matrices are equivalent to linear maps Rⁿ → R^m, as described above. More generally, any linear map f: V → W between finite-dimensional vector spaces can be described by a matrix A = (a_ij), by choosing bases v₁, ..., v_n, and w₁, ..., w_m, where n and m are the dimensions of V and W, respectively, and requiring

$f(v_i) = \sum_{j=1}^m a_{i,j} w_j$ .

This uniquely determines the entries of the matrix A, but the matrix depends on the choice of the bases: different choices of bases give rise to different, but similar matrices. Many of the above concrete notions can be reinterpreted in this light, for example, the transpose matrix A^T describes the transpose of the linear map given by A, with respect to the dual bases.

It is also possible to consider matrices with infinitely many rows and/or columns. The basic operations introduced above are defined in the same way in this case. Matrix multiplication, however, and all operations stemming therefrom are only meaningful when restricted to certain matrices, since the sum featuring in the above definition of the matrix product will contain an infinity of summands. An easy way to circumvent this issue is to restrict to matrices all of whose rows (or columns) contain only finitely many nonzero terms. More generally, one can impose convergence restrictions on the entries of the matrices involved. The explicit point of view of matrices tends to obfuscate the matter, and the abstract and more powerful tools of functional analysis are used instead, by relating matrices to linear maps (as in the finite case above), but imposing additional convergence and continuity constraints.

Another step of generalization is matrices with entries in a ring R.^[26] Rings are a more general notion than fields in that no division operation exists. The very same addition and multiplication operations of matrices extend to this setting, too. The set M(n, R) of all square n-by-n matrices over R is a ring in its own right, isomorphic to the endomorphism ring of the left R-module Rⁿ. If the ring R is commutative, i.e. its multiplication is commutative, then M(n, R) is a unitary noncommutative (unless n = 1) associative algebra over R. The determinant of square matrices can still be defined using the Leibniz formula; a matrix is invertible if and only if its determinant is invertible in R, generalising the situation over a field F, where every nonzero element is invertible. Matrices over superrings are called supermatrices.

Matrix groups

Main article: Matrix group

A group is a mathematical structure consisting of a set of objects together with a binary operation, i.e. an operation combining any two objects to a third, subject to certain requirements. A group in which the objects are matrices and the group operation is matrix multiplication is called a matrix group. Since in a group every element has to be invertible, the most general matrix groups are the groups of all invertible matrices of a given size, called the general linear groups.

Any property of matrices that is preserved under matrix products and inverses can be used to define further matrix groups. For example, matrices with a given size and with a determinant of 1 form a subgroup of (i.e. a smaller group contained in) their general linear group, called a special linear group. Orthogonal matrices, determined by the condition

M^TM = I

form the orthogonal group.^[27] They are called orthogonal since the associated linear transformations of Rⁿ preserve angles in the sense that the scalar product of two vectors is unchanged after applying M to them:

(Mv) · (Mw) = v · w.^[28]

Every finite group is isomorphic to a matrix group. General groups can be studied using matrix groups, which are comparatively well-understood, by means of representation theory.

Applications

There are manifold applications of matrices, both in mathematics and other sciences. Some of them merely take advantage of the compact representation of a set of numbers in a matrix. For example, in game theory, the payoff matrix encodes the payoff for two players, depending on which out of a given (finite) set of alternatives the players chose. Text mining and automatated thesaurus compilation makes use of document-term matrices such as tf-idf in order to keep track of frequencies of certain words in several documents.

Complex numbers can be represented by particular real 2-by-2 matrices via

$a + ib \leftrightarrow \begin{bmatrix} a & -b \\ b & a \end{bmatrix}$ ,

under wich addition and multiplication of complex numbers and matrices correspond to each other. For example, 2-by-2 rotation matrices represent the multiplication with some complex number of absolute value 1, as above. A similar interpretation is possible for quaternions.

Matrices over a polynomial ring are important in the study of control theory.

Early encryption techniques such as the Hill cipher also used matrices. However, due to the linear nature of matrices, these codes are comparatively easy to break.

Graph theory

An undirected graph with adjacency matrix $\begin{bmatrix} 1 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}$ .

The adjacency matrix of a finite graph is a basic notion of graph theory. It saves which vertices of the graph are connected by an edge. Matrices containing just two different values (0 and 1 meaning for example "yes" and "no") are called logical matrices. The distance matrix contains information about the distance of the edges. This can be applied to websites connected hyperlinks or cities connected by roads etc., in which case (unless the road network is extremely dense) the matrices tend to be sparse, i.e. contain few nonzero entries. Therefore, specifically taylored matrix algorithms can be used in network theory.

Analysis and geometry

At the saddle point (red) of the function f(x, y) = x² − y², the Hessian matrix is indefinite.

The Hessian matrix of a differentiable function f: Rⁿ → R consists of the second derivatives of f with respect to the several coordinate directions, i.e.

$H(f) = \left [\frac {\partial^2f}{\partial x_i \partial x_j} \right ]$

It encodes information about the local growth behaviour of the function: if it is positive definite at some point, i.e. all eigenvalues are positive, the function has a local minimum there. Quadratic programming can be used to find global minima or maxima of quadratic functions closely related to the ones attached to matrices (see above).

Another matrix frequently used in geometrical situations is the Jacobi matrix of a differentiable map f: Rⁿ → R^m. If the components f₁, ..., f_m denote the components of f, then the Jacobi matrix is defined as

$J(f) = \left [\frac {\partial f_i}{\partial x_j} \right ]_{1 \leq i \leq m, 1 \leq j \leq n}.$

If n > m, and if the rank of the Jacobi matrix is maximal, i.e. m, the function is locally invertible at that point, by the implicit function theorem.

Partial differential equations can be classified by considering the matrix of coefficients of the highest-order differential operators of the equation, for elliptic partial differential equations this matrix is positive definite, which has decisive influence on the set of possible solutions of the equation in question.

The finite element method is an important numerical method to solve partial differential equations, widely applied in simulating complex physical systems. It attempts to approximate the solution to some equation by piecewise linear functions, where the pieces are chosen with respect to a sufficiently fine grid, which in turn can be recast as a matrix equation.

Probability theory and statistics

Two different Markov chains. The chart depicts the number of particles (of a total of 1000) in state "2".

Stochastic matrices are square matrices whose rows are probability vectors, i.e. whose entries sum up to one. Stochastic matrices are used to define Markov chains with finitely many states.^[29] In the image at the right, particles can attain two states ("1" or "2"), the stochastic matrices give the probability distribution for the next position of some particle, depending on its current position. In the example at the right, particles move randomly between the two states, initially they are all in state "1" (red chain, starting from below) and state "2" (black, respectively), the transition matrices are given by $\begin{pmatrix}.7&0\\.3&1\end{pmatrix}$ and $\begin{pmatrix}.7&.2\\.3&.8\end{pmatrix}$ , respectively, e.g. the probability for a particle in state "2" to stay in state "2" is 1 in the first example. Therefore this state is absorbing, i.e. all particles will attain this state eventually. This property, and the limiting value in the other case can be read of from the eigenvectors of the transition matrices.

The covariance matrix encodes the mutual variance of several random variables.

Symmetries and transformations in physics

Further information: Symmetry in physics

Linear transformation and the associated symmetries play a key role in modern physics. For example, elementary particles in quantum field theory are classified as representations of the Lorentz group of special relativity and, more specifically, by their behavior under the spin group. Concrete representations involving the Pauli matrices and more general gamma matrices are an integral part of the physical description of fermions, which behave as spinors.^[30] For the three lightest quarks, there is a group-theoretical representation involving the special unitary group SU(3); for their calculations, physicists use a convenient matrix representation known as the Gell-Mann matrices, which are also used for the SU(3) gauge group that forms the basis of the mordern description of strong nuclear interactions, quantum chromodynamics. The Cabibbo-Kobayashi-Maskawa matrix, in turn, expresses the fact that the basic quark states that are important for weak interactions are not the same as, but linearly related to the basic quark states that define particles with specific and distinct masses.^[31]

Linear combinations of quantum states

The first model of quantum mechanics (Heisenberg, 1925) represented the theory's operators by infinite-dimensional matrices acting on quantum states.^[32] This is also referred to as matrix mechanics. One particular example is the density matrix that characterizes the "mixed" state of a quantum system as a linear combination of elementary, "pure" eigenstates.^[33]

Another matrix serves as a key tool for describing the scattering experiments which form the cornerstone of experimental particle physics: Collision reactions such as occur in particle accelerators, where non-interacting particles head towards each other and collide in a small interaction zone, with a new set of non-interacting particles as the result, can be described as the scalar product of outgoing particle states and a linear combination of ingoing particle states. The linear combination is given by a matrix known as the S-matrix, which encodes all information about the possible interactions between particles.^[34]

Normal Modes

A general application of matrices in physics is to the description of linearly coupled harmonic systems. The equations of motion of such systems can be described in matrix form, with a mass matrix multiplying a generalized velocity to give the kinetic term, and a force matrix multiplying a displacement vector to characterize the interactions. The best way to obtain solutions is to determine the system's eigenvectors, its normal modes, by diagonalizing the matrix equation. Techniques like this are crucial when it comes to describing the internal dynamics of molecules: the internal vibrations of systems consisting of mutually bound component atoms.^[35] They are also needed for describing mechanical vibrations, and oscillations in electrical circuits.^[36]

Geometrical optics

Geometrical optics provides further matrix applications. In this approximative theory, the wave nature of light is neglected. The result is a model in which light rays are indeed geometrical rays. If the deflection of light rays by optical elements is small, the action of a lens or reflective element on a given light ray can be expressed as multiplication of a two-component vector with a two-by-two matrix called ray transfer matrix: the vector's components are the light ray's slope and its distance from the optical axis, while the matrix encodes the properties of the optical element. The matrix characterizing an optical system consisting of a combination of lenses and/or reflective elements is simply the product of the components' matrices.^[37]

Computer graphics

4×4 transformation affine rotation matrices are commonly used in computer graphics. A 4x4 matrix allows for the simple representation of arbitrary combinations of affine transformations and projective transforms as well as translations. This allows for the transformation of arbitraily positioned objects in a 3D scene into a 2D image, corresponding to a theoretical camera observation.

The upper left 3×3 portion of a transformation matrix is composed of the new X, Y, and Z axes of the post-transformation coordinate space.

Footnotes

^ Brown 1991, Chapter I.1.
^ ^a ^b Swaney, Mark. History of Magic Squares.
^ Shen Kangshen et al. (ed.) (1999). Nine Chapters of the Mathematical Art, Companion and Commentary. Oxford University Press. cited by Otto Bretscher (2005). Linear Algebra with Applications (3rd ed. ed.). Prentice-Hall. pp. p. 1.
^ Brown 1991, Definition I.2.1.
^ Brown 1991, Definition I.2.4.
^ Brown 1991, Definition I.2.33.
^ Brown 1991, Theorem I.2.6.
^ Brown 1991, Definition I.2.20.
^ Brown 1991, Theorem I.2.24.
^ Brown 1991, I.2.21 and 22.
^ Brown 1991, Definition II.3.3.
^ Brown 1991, Theorem II.3.22.
^ Brown 1991, Definition I.2.28.
^ Brown 1991, Definition I.5.13.
^ Brown 1991, Definition III.2.1.
^ Brown 1991, Corollary III.2.16.
^ Brown 1991, Theorem III.2.12.
^ Brown 1991, Theorem III.3.18.
^ Brown 1991, Definition III.4.1.
^ Brown 1991, Definition III.4.9.
^ Brown 1991, Corollary III.4.10.
^ Bau III & Trefethen 1997.
^ Golub & Van Loan 1996, Algorithm 1.3.1.
^ Golub & Van Loan 1996, Chapter 2.3.
^ Golub & Van Loan 1996, Chapters 9 and 10, esp. section 10.2.
^ Lang 2002, Chapter XIII.
^ Artin 1991, Chapter 4.5.
^ Artin 1991, Theorem 4.5.13.
^ Latouche & Ramaswami 1999.
^ Itzykson & Zuber 1980, Ch. 2.
^ see Burgess & Moore 2007, section 1.6.3. (SU(3)), section 2.4.3.2. (Kobayashi-Maskawa matrix).
^ Schiff 1968, Ch. 6.
^ Bohm 2001, sections II.4 and II.8.
^ Weinberg 1995, Ch. 3.
^ Wherrett 1987, part II.
^ Riley, Hobson & Bence 1997, 7.17.
^ Guenther 1990, Ch. 5.

^ Eigen means "own" in German and in Dutch.

References

Artin, Michael (1991), Algebra, Prentice Hall, ISBN 978-0-89871-510-1
Bau III, David; Trefethen, Lloyd N. (1997), Numerical linear algebra, Philadelphia: Society for Industrial and Applied Mathematics, ISBN 978-0-89871-361-9
Brown, William A. (1991), Matrices and vector spaces, New York: M. Dekker, ISBN 978-0-8247-8419-5
Golub, Gene H.; Van Loan, Charles F. (1996), Matrix Computations (3rd ed.), Johns Hopkins, ISBN 978-0-8018-5414-9
Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, 211, Berlin, New York: Springer-Verlag, MR 1878556, ISBN 978-0-387-95385-4
Latouche, G.; Ramaswami, V. (1999), Introduction to matrix analytic methods in stochastic modeling (1st ed.), Philadelphia: Society for Industrial and Applied Mathematics, ISBN 978-0-89871-425-8

Physics references

Bohm, Arno (2001), Quantum Mechanics: Foundations and Applications, Springer, ISBN 0-387-95330-2
Burgess, Cliff; Moore, Guy (2007), The Standard Model. A Primer, Cambridge University Press, ISBN 0-521-86036-9
Guenther, Robert D. (1990), Modern Optics, John Wiley, ISBN 0-471-60538-7
Itzykson, Claude; Zuber, Jean-Bernard (1980), Quantum Field Theory, McGraw-Hill, ISBN 0-07-032071-3
Riley, K. F.; Hobson, M. P.; Bence, S. J. (1997), Mathematical methods for physics and engineering, Cambridge University Press, ISBN 0-521-55506-X
Schiff, Leonard I. (1968), Quantum Mechanics (3rd ed.), McGraw-Hill
Weinberg, Steven (1995), The Quantum Theory of Fields. Volume I: Foundations, Cambridge University Press, ISBN 0-521-55001-7
Wherrett, Brian S. (1987), Group Theory for Atoms, Molecules and Solids, Prentice-Hall International, ISBN 0-13-365461-3

External links

Online books

Kaw, Autar K., Introduction to Matrix Algebra, ISBN 978-0-615-25126-4, http://autarkaw.com/books/matrixalgebra/index.html
The Matrix Cookbook, http://matrixcookbook.com, retrieved on 10 December 2008
Brookes, M. (2005), The Matrix Reference Manual, London: Imperial College, http://www.ee.ic.ac.uk/hp/staff/dmb/matrix/intro.html, retrieved on 10 December 2008

Online Matrix Calculators

Xiao, Gang, Matrix calculator, http://wims.unice.fr/wims/wims.cgi?module=tool/linear/matrix.en, retrieved on 10 December 2008
Online matrix calculator, http://www.bluebit.gr/matrix-calculator/, retrieved on 10 December 2008

Oehlert, Gary W.; Bingham, Christopher, MacAnova, University of Minnesota, School of Statistics, http://www.stat.umn.edu/macanova/macanova.home.html, retrieved on 10 December 2008 , a freeware package for matrix algebra and statistics

[0] Brown 1991, Chapter I.1.

[Swaney-1] Swaney, Mark. History of Magic Squares.

[2] Shen Kangshen et al. (ed.) (1999). Nine Chapters of the Mathematical Art, Companion and Commentary. Oxford University Press. cited by Otto Bretscher (2005). Linear Algebra with Applications (3rd ed. ed.). Prentice-Hall. pp. p. 1.

[3] Brown 1991, Definition I.2.1.

[4] Brown 1991, Definition I.2.4.

[5] Brown 1991, Definition I.2.33.

[6] Brown 1991, Theorem I.2.6.

[7] Brown 1991, Definition I.2.20.

[8] Brown 1991, Theorem I.2.24.

[9] Brown 1991, I.2.21 and 22.

[10] Brown 1991, Definition II.3.3.

[11] Brown 1991, Theorem II.3.22.

[12] Brown 1991, Definition I.2.28.

[13] Brown 1991, Definition I.5.13.

[14] Brown 1991, Definition III.2.1.

[15] Brown 1991, Corollary III.2.16.

[16] Brown 1991, Theorem III.2.12.

[17] Brown 1991, Theorem III.3.18.

[19] Brown 1991, Definition III.4.1.

[20] Brown 1991, Definition III.4.9.

[21] Brown 1991, Corollary III.4.10.

[22] Bau III & Trefethen 1997.

[23] Golub & Van Loan 1996, Algorithm 1.3.1.

[24] Golub & Van Loan 1996, Chapter 2.3.

[25] Golub & Van Loan 1996, Chapters 9 and 10, esp. section 10.2.

[26] Lang 2002, Chapter XIII.

[27] Artin 1991, Chapter 4.5.

[28] Artin 1991, Theorem 4.5.13.

[29] Latouche & Ramaswami 1999.

[30] Itzykson & Zuber 1980, Ch. 2.

[31] see Burgess & Moore 2007, section 1.6.3. (SU(3)), section 2.4.3.2. (Kobayashi-Maskawa matrix).

[32] Schiff 1968, Ch. 6.

[33] Bohm 2001, sections II.4 and II.8.

[34] Weinberg 1995, Ch. 3.

[35] Wherrett 1987, part II.

[36] Riley, Hobson & Bence 1997, 7.17.

[37] Guenther 1990, Ch. 5.

[18] Eigen means "own" in German and in Dutch.

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