Matrix (mathematics)

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

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, $det (λ I - A)$ .

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.

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.Template:Sfnp Most commonly, a matrix over a field $F$ is a rectangular array of elements of Template:Tmath.Template:Sfnp Template:Sfnp 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: $𝐀 = [\begin{matrix} - 1.3 & 0.6 \\ 20.4 & 5.5 \\ 9.7 & - 6.2 \end{matrix}] .$

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.Template:Sfnp

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 Template:Mvar rows and Template:Mvar columns is called an Template:Math matrix,Template:Sfnp or Template:Mvar-by-Template:Mvar matrix,Template:Sfnp where Template:Mvar and Template:Mvar are called its Template:Dfni.Template:Sfnp For example, the matrix $𝐀$ above is a Template:Math 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.Template:Sfnp 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.^[1]

Overview of a matrix size
Name	Size	Example	Description
Row matrix	$1 \times n$	$[\begin{matrix} 3 & 7 & 2 \end{matrix}]$	A matrix with one row and more than one columns, sometimes used to represent a vector
Column matrix	$n \times 1$	$[\begin{matrix} 4 \\ 1 \\ 8 \end{matrix}]$	A matrix with one column and more than one rows, sometimes used to represent a vector
Square matrix	$n \times n$	$[\begin{matrix} 9 & 13 & 5 \\ 1 & 11 & 7 \\ 2 & 6 & 3 \end{matrix}]$	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,Template:Sfnp so that an Template:Math matrix $𝐀$ is represented as $𝐀 = [\begin{matrix} a_{11} & a_{12} & \dots & a_{1 n} \\ a_{21} & a_{22} & \dots & a_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ a_{m 1} & a_{m 2} & \dots & a_{m n} \end{matrix}] = (\begin{matrix} a_{11} & a_{12} & \dots & a_{1 n} \\ a_{21} & a_{22} & \dots & a_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ a_{m 1} & a_{m 2} & \dots & a_{m n} \end{matrix}) .$ This may be abbreviated by writing only a single generic term, possibly along with indices, as in $𝐀 = (a_{i j}), [a_{i j}], or {(a_{i j})}_{1 \leq i \leq m, 1 \leq j \leq n}$ or $𝐀 = (a_{i, j})_{1 \leq i, j \leq n}$ in the case that Template:Tmath.

Matrices are usually symbolized using upper-case letters (such as $𝐀$ in the examples above),Template:Sfnp while the corresponding lower-case letters, with two subscript indices (e.g., Template:Tmath, or Template:Tmath), represent the entries.Template:Sfnp 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 Template:Tmath.Template:Sfnp

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

Sometimes, the entries of a matrix can be defined by a formula such as Template:Tmath. For example, each of the entries of the following matrix $𝐀$ is determined by the formula Template:Tmath. $𝐀 = [\begin{matrix} 0 & - 1 & - 2 & - 3 \\ 1 & 0 & - 1 & - 2 \\ 2 & 1 & 0 & - 1 \end{matrix}]$ 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 Template:Tmath. If matrix size is Template:Math, the above-mentioned formula $f (i, j)$ is valid for any $i = 1, \dots, m$ and any Template:Tmath. This can be specified separately or indicated using Template:Math as a subscript. For instance, the matrix $𝐀$ above is Template:Math, and can be defined as $𝐀 = [i - j] (i = 1, 2, 3; j = 1, \dots, 4)$ or Template:Tmath.

Some programming languages utilize doubly subscripted arrays (or arrays of arrays) to represent an {Template:Mvar-by-Template:Mvar matrix. Some programming languages start the numbering of array indexes at zero, in which case the entries of an Template:Math matrix are indexed by $0 \leq i \leq m - 1$ and Template:Tmath.Template:Sfnp This article follows the more common convention in mathematical writing where enumeration starts from Template:Math.

The set of all Template:Mvar-by-Template:Mvar real matrices is often denoted Template:Tmath, or Template:Tmath. The set of all Template:Math matrices over another field, or over a ring Template:Math, is similarly denoted Template:Tmath, or Template:Tmath. If Template:Math, such as in the case of square matrices, one does not repeat the dimension: Template:Tmath, or Template:Tmath.Template:Sfnp Often, Template:Tmath, or Template:Tmath, is used in place of Template:Tmath.^[2]

Basic operations

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.Template:Sfnp

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

Addition

Template:Main

Matrix addition and subtraction require matrices of a consistent size, and are calculated entrywise. The sum Template:Math and the difference Template:Math of two Template:Math matrices are:Template:Sfnp

$\begin{matrix} (𝐀 + 𝐁)_{i, j} = 𝐀_{i, j} + 𝐁_{i, j}, 1 \leq i \leq m, 1 \leq j \leq n . \\ (𝐀 - 𝐁)_{i, j} = 𝐀_{i, j} - 𝐁_{i, j}, 1 \leq i \leq m, 1 \leq j \leq n . \end{matrix}$

For example, $[\begin{matrix} 1 & 3 & 1 \\ 1 & 0 & 0 \end{matrix}] + [\begin{matrix} 0 & 0 & 5 \\ 7 & 5 & 0 \end{matrix}] = [\begin{matrix} 1 + 0 & 3 + 0 & 1 + 5 \\ 1 + 7 & 0 + 5 & 0 + 0 \end{matrix}] = [\begin{matrix} 1 & 3 & 6 \\ 8 & 5 & 0 \end{matrix}]$ Template:Pb

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: Template:Math.Template:Sfnp

Scalar multiplication

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The product Template:Math of a number Template:Mvar (also called a scalar in this context) and a matrix Template:Math is computed by multiplying each entry of Template:Math by Template:Mvar:Template:Sfnp $(c 𝐀)_{i, j} = c \cdot 𝐀_{i, j}$ 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".Template:Sfnp For example: $2 \cdot [\begin{matrix} 1 & 8 & - 3 \\ 4 & - 2 & 5 \end{matrix}] = [\begin{matrix} 2 \cdot 1 & 2 \cdot 8 & 2 \cdot - 3 \\ 2 \cdot 4 & 2 \cdot - 2 & 2 \cdot 5 \end{matrix}] = [\begin{matrix} 2 & 16 & - 6 \\ 8 & - 4 & 10 \end{matrix}]$ Template:Pb

Matrix subtraction is consistent with composition of matrix addition with scalar multiplication by Template:Math:Template:Sfnp $𝐀 - 𝐁 = 𝐀 + (- 1) \cdot 𝐁$ Template:Pb

Transpose

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

The transpose is compatible with addition and scalar multiplication, as expressed by Template:Math and Template:Math. Finally, Template:Math.Template:Sfnp

Matrix multiplication

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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 Template:Math is an Template:Math matrix and Template:Math is an Template:Math matrix, then their matrix product Template:Math is the Template:Math matrix whose entries are given by the dot product of the corresponding row of Template:Math and the corresponding column of Template:Math:Template:Sfnp $[𝐀 𝐁]_{i, j} = a_{i, 1} b_{1, j} + a_{i, 2} b_{2, j} + \dots + a_{i, n} b_{n, j} = \sum_{r = 1}^{n} a_{i, r} b_{r, j},$ where Template:Math and Template:Math.Template:Sfnp For example, the underlined entry 2340 in the product is calculated as Template:Math $\begin{matrix} [\begin{matrix} \underline{2} & \underline{3} & \underline{4} \\ 1 & 0 & 0 \end{matrix}] [\begin{matrix} 0 & \underline{1000} \\ 1 & \underline{100} \\ 0 & \underline{10} \end{matrix}] & = [\begin{matrix} 3 & \underline{2340} \\ 0 & 1000 \end{matrix}] . \end{matrix}$

Matrix multiplication satisfies the rules Template:Math (associativity), and Template:Math as well as Template:Math (left and right distributivity), whenever the size of the matrices is such that the various products are defined.Template:Sfnp The product Template:Math may be defined without Template:Math being defined, namely if Template:Math and Template:Math are Template:Math and Template:Math matrices, respectively, and Template:Math Even if both products are defined, they generally need not be equal, that is:Template:Sfnp $𝐀 𝐁 \neq 𝐁 𝐀 .$

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.Template:Sfnp An example of two matrices not commuting with each other is: $[\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}] [\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}] = [\begin{matrix} 0 & 1 \\ 0 & 3 \end{matrix}],$ whereas $[\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}] [\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}] = [\begin{matrix} 3 & 4 \\ 0 & 0 \end{matrix}] .$

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.Template:Sfnp They arise in solving matrix equations such as the Sylvester equation.Template:Sfnp

Row operations

Template:Main There are three types of row operations:Template:Sfnp Template:Sfnp

row addition, that is, adding a row to another.
row multiplication, that is, multiplying all entries of a row by a non-zero constant;
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.Template:Sfnp

Submatrix

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

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

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.Template:Sfnp Template:Sfnp Other authors define a principal submatrix as one in which the first Template:Mvar rows and columns, for some number Template:Mvar, are the ones that remain;Template:Sfnp this type of submatrix has also been called a leading principal submatrix.^[3]

Linear equations

Template:Main Matrices can be used to compactly write and work with multiple linear equations, that is, systems of linear equations. For example, if Template:Math is an Template:Math matrix, Template:Math designates a column vector (that is, Template:Math matrix) of Template:Mvar variables Template:Math and Template:Math is an Template:Math column vector, then the matrix equation $𝐀 𝐱 = 𝐛$ is equivalent to the system of linear equationsTemplate:Sfnp $\begin{matrix} a_{1, 1} x_{1} + a_{1, 2} x_{2} + & \dots + a_{1, n} x_{n} = b_{1} \\ ⋮ \\ a_{m, 1} x_{1} + a_{m, 2} x_{2} + & \dots + a_{m, n} x_{n} = b_{m} \end{matrix}$

Using matrices, this can be solved more compactly than would be possible by writing out all the equations separately. If Template:Math and the equations are independent, then this can be done by writingTemplate:Sfnp $𝐱 = 𝐀^{- 1} 𝐛$ where Template:Math is the inverse matrix of Template:Math. If Template:Math has no inverse, solutions—if any—can be found using its generalized inverse.Template:Sfnp

Linear transformations

Template:Main

The vectors represented by a Template:Math matrix correspond to the sides of a unit square transformed into a parallelogram.

Matrices and matrix multiplication reveal their essential features when related to linear transformations, also known as linear maps. A real Template:Mvar-by-Template:Mvar matrix Template:Math gives rise to a linear transformation $ℝ^{n} \to ℝ^{m}$ mapping each vector Template:Math in Template:Tmath to the (matrix) product Template:Math, which is a vector in Template:Tmath Conversely, each linear transformation $f : ℝ^{n} \to ℝ^{m}$ arises from a unique Template:Mvar-by-Template:Mvar matrix Template:Math: explicitly, the Template:Math-entry of Template:Math is the Template:Mvarth coordinate of Template:Math, where Template:Math is the unit vector with Template:Math in the Template:Mvarth position and Template:Math elsewhere. The matrix Template:Math is said to represent the linear map Template:Mvar, and Template:Math is called the transformation matrix of Template:Mvar.Template:Sfnp

For example, the Template:Math matrix $𝐀 = [\begin{matrix} a & c \\ b & d \end{matrix}]$ can be viewed as the transform of the unit square into a parallelogram with vertices at Template:Math, Template:Math, Template:Math, and Template:Math. The parallelogram pictured at the right is obtained by multiplying Template:Math with each of the column vectors Template:Tmath, Template:Tmath, Template:Tmath, and Template:Tmath in turn. These vectors define the vertices of the unit square.Template:Sfnp The following table shows several Template:Math real matrices with the associated linear maps of Template:Tmath. The Template:Font color original is mapped to the Template:Font color grid and shapes. The origin Template:Math is marked with a black point.

Horizontal shear Template:Sfnp Template:Brwith m = 1.25.	Reflection Template:Sfnp through the vertical axis	Squeeze mapping Template:Sfnp Template:Brwith r = 3/2	Scaling Template:Sfnp Template:Brby a factor of 3/2	Rotation Template:Sfnp Template:Brby Template:Pi/6 = 30°
$[\begin{matrix} 1 & 1.25 \\ 0 & 1 \end{matrix}]$	$[\begin{matrix} - 1 & 0 \\ 0 & 1 \end{matrix}]$	$[\begin{matrix} \frac{3}{2} & 0 \\ 0 & \frac{2}{3} \end{matrix}]$	$[\begin{matrix} \frac{3}{2} & 0 \\ 0 & \frac{3}{2} \end{matrix}]$	$[\begin{matrix} \cos (\frac{π}{6}) & - \sin (\frac{π}{6}) \\ \sin (\frac{π}{6}) & \cos (\frac{π}{6}) \end{matrix}]$

Under the 1-to-1 correspondence between matrices and linear maps, matrix multiplication corresponds to composition of maps:^[4] if a Template:Mvar-by-Template:Mvar matrix Template:Math represents another linear map Template:Tmath, then the composition Template:Math is represented by Template:Math sinceTemplate:Sfnp $(g \circ f) (𝐱) = g (f (𝐱)) = g (𝐀 𝐱) = 𝐁 (𝐀 𝐱) = (𝐁 𝐀) 𝐱 .$

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

The rank of a matrix Template:Math 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.Template:Sfnp Equivalently it is the dimension of the image of the linear map represented by Template:Math.Template:Sfnp 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.Template:Sfnp

Square matrix

Template:Main A square matrix is a matrix with the same number of rows and columns. An Template:Mvar-by-Template:Mvar matrix is known as a square matrix of order Template:Mvar. Any two square matrices of the same order can be added and multiplied. The entries Template:Mvar 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.Template:Sfnp

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

Main types

Name	Example with Template:Math
Diagonal matrix	$[\begin{matrix} a_{11} & 0 & 0 \\ 0 & a_{22} & 0 \\ 0 & 0 & a_{33} \end{matrix}]$
Lower triangular matrix	$[\begin{matrix} a_{11} & 0 & 0 \\ a_{21} & a_{22} & 0 \\ a_{31} & a_{32} & a_{33} \end{matrix}]$
Upper triangular matrix	$[\begin{matrix} a_{11} & a_{12} & a_{13} \\ 0 & a_{22} & a_{23} \\ 0 & 0 & a_{33} \end{matrix}]$

Diagonal and triangular matrix

If all entries of Template:Math below the main diagonal are zero, Template:Math is called an upper triangular matrix. Similarly, if all entries of Template:Math above the main diagonal are zero, Template:Math is called a lower triangular matrix.Template:Sfnp If all entries outside the main diagonal are zero, Template:Math is called a diagonal matrix.Template:Sfnp

Identity matrix

Template:Main The identity matrix Template:Math of size Template:Mvar is the Template:Mvar-by-Template:Mvar matrix in which all the elements on the main diagonal are equal to Template:Math and all other elements are equal to Template:Math,Template:Sfnp for example, $\begin{matrix} 𝐈_{1} & = [\begin{matrix} 1 \end{matrix}], \\ 𝐈_{2} & = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}], \\ ⋮ \\ 𝐈_{n} & = [\begin{matrix} 1 & 0 & \dots & 0 \\ 0 & 1 & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & 1 \end{matrix}] \end{matrix}$ It is a square matrix of order Template:Mvar, and also a special kind of diagonal matrix. It is called an identity matrix because multiplication with it leaves a matrix unchanged:Template:Sfnp ${𝐀 𝐈}_{n} = 𝐈_{m} 𝐀 = 𝐀$ for any Template:Mvar-by-Template:Mvar matrix Template:Math.

A scalar multiple of an identity matrix is called a scalar matrix.Template:Sfnp

Symmetric or skew-symmetric matrix

A square matrix Template:Math that is equal to its transpose, that is, Template:Math, is a symmetric matrix. If instead, Template:Math is equal to the negative of its transpose, that is, Template:Math, then Template:Math is a skew-symmetric matrix. In complex matrices, symmetry is often replaced by the concept of Hermitian matrices, which satisfies Template:Math, where the star or asterisk denotes the conjugate transpose of the matrix, that is, the transpose of the complex conjugate of Template:Math.Template:Sfnp

By the spectral theorem, real symmetric matrices and complex Hermitian matrices have an eigenbasis; that is, every vector is expressible as a linear combination of eigenvectors. In both cases, all eigenvalues are real.Template:Sfnp This theorem can be generalized to infinite-dimensional situations related to matrices with infinitely many rows and columns.Template:Sfnp

Invertible matrix and its inverse

A square matrix Template:Math is called invertible or non-singular if there exists a matrix Template:Math such thatTemplate:Sfnp Template:Sfnp $𝐀 𝐁 = 𝐁 𝐀 = 𝐈_{n},$ where Template:Math is the Template:Math identity matrix with Template:Math for each entry on the main diagonal and Template:Math elsewhere. If Template:Math exists, it is unique and is called the inverse matrix of Template:Math, denoted Template:Math.Template:Sfnp

There are many algorithms for testing whether a square matrix is invertible, and, if it is, computing its inverse. One of the oldest, which is still in common use is Gaussian elimination.Template:Sfnp

Definite matrix

Positive definite matrix	Indefinite matrix
$[\begin{matrix} \frac{1}{4} & 0 \\ 0 & 1 \end{matrix}]$	$[\begin{matrix} \frac{1}{4} & 0 \\ 0 & - \frac{1}{4} \end{matrix}]$
$Q (x, y) = \frac{1}{4} x^{2} + y^{2}$	$Q (x, y) = \frac{1}{4} x^{2} - \frac{1}{4} y^{2}$
Template:BrPoints such that $Q (x, y) = 1$ Template:Br (Ellipse)	Template:Br Points such that $Q (x, y) = 1$ Template:Br (Hyperbola)

A symmetric real matrix Template:Math is called positive-definite if the associated quadratic form $f (𝐱) = 𝐱^{T} 𝐀 𝐱$ has a positive value for every nonzero vector Template:Math in Template:Tmath. If Template:Math yields only negative values then Template:Math is negative-definite; if Template:Mvar does produce both negative and positive values then Template:Math is indefinite.Template:Sfnp If the quadratic form Template:Mvar yields only non-negative values (positive or zero), the symmetric matrix is called positive-semidefinite (or if only non-positive values, then negative-semidefinite); hence the matrix is indefinite precisely when it is neither positive-semidefinite nor negative-semidefinite.Template:Sfnp

A symmetric matrix is positive-definite if and only if all its eigenvalues are positive, that is, the matrix is positive-semidefinite and it is invertible.Template:Sfnp The table at the right shows two possibilities for 2-by-2 matrices. The eigenvalues of a diagonal matrix are simply the entries along the diagonal,Template:Sfnp and so in these examples, the eigenvalues can be read directly from the matrices themselves. The first matrix has two eigenvalues that are both positive, while the second has one that is positive and another that is negative.

Allowing as input two different vectors instead yields the bilinear form associated to Template:Math:Template:Sfnp $B_{𝐀} (𝐱, 𝐲) = 𝐱^{T} 𝐀 𝐲 .$

In the case of complex matrices, the same terminology and results apply, with symmetric matrix, quadratic form, bilinear form, and transpose Template:Math replaced respectively by Hermitian matrix, Hermitian form, sesquilinear form, and conjugate transpose Template:Math.Template:Sfnp

Orthogonal matrix

Template:Main An orthogonal matrix is a square matrix with real entries whose columns and rows are orthogonal unit vectors (that is, orthonormal vectors).Template:Sfnp Equivalently, a matrix Template:Math is orthogonal if its transpose is equal to its inverse: $𝐀^{T} = 𝐀^{- 1},$ which entails $𝐀^{T} 𝐀 = 𝐀 𝐀^{T} = 𝐈_{n},$ where Template:Math is the identity matrix of size Template:Mvar.Template:Sfnp

An orthogonal matrix Template:Math is necessarily invertible (with inverse Template:Math), unitary (Template:Math), and normal (Template:Math). The determinant of any orthogonal matrix is either Template:Math or Template:Math. A special orthogonal matrix is an orthogonal matrix with determinant Template:Math. As a linear transformation, every orthogonal matrix with determinant Template:Math is a pure rotation without reflection, i.e., the transformation preserves the orientation of the transformed structure, while every orthogonal matrix with determinant Template:Math reverses the orientation, i.e., is a composition of a pure reflection and a (possibly null) rotation. The identity matrices have determinant Template:Math and are pure rotations by an angle zero.Template:Sfnp

The complex analog of an orthogonal matrix is a unitary matrix.Template:Sfnp

Main operations

Trace

The trace, Template:Math of a square matrix Template:Math is the sum of its diagonal entries. While matrix multiplication is not commutative as mentioned above, the trace of the product of two matrices is independent of the order of the factors:Template:Sfnp $tr (𝐀 𝐁) = tr (𝐁 𝐀) .$ This is immediate from the definition of matrix multiplication:Template:Sfnp $tr (𝐀 𝐁) = \sum_{i = 1}^{m} \sum_{j = 1}^{n} a_{i j} b_{j i} = tr (𝐁 𝐀) .$ It follows that the trace of the product of more than two matrices is independent of cyclic permutations of the matrices; however, this does not in general apply for arbitrary permutations. For example, Template:Math, in general.Template:Sfnp Also, the trace of a matrix is equal to that of its transpose,Template:Sfnp that is, $tr (𝐀) = tr (𝐀^{T}) .$

Determinant

Template:Main

The determinant of a square matrix Template:Math (denoted Template:Math or Template:Math) is a number encoding certain properties of the matrix. A matrix is invertible if and only if its determinant is nonzero.Template:Sfnp Its absolute value equals the area (in Template:Tmath) or volume (in Template:Tmath) 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.Template:Sfnp

The determinant of Template:Math matrices is given by^[5] $det [\begin{matrix} a & b \\ c & d \end{matrix}] = a d - b c .$ The determinant of Template:Math matrices involves six terms (rule of Sarrus). The more lengthy Leibniz formula generalizes these two formulae to all dimensions.Template:Sfnp

The determinant of a product of square matrices equals the product of their determinants: $det (𝐀 𝐁) = det (𝐀) \cdot det (𝐁),$ or using alternate notation:Template:Sfnp $| 𝐀 𝐁 | = | 𝐀 | \cdot | 𝐁 | .$ Adding a multiple of any row to another row, or a multiple of any column to another column, does not change the determinant. Interchanging two rows or two columns affects the determinant by multiplying it by Template:Math.Template:Sfnp Using these operations, any matrix can be transformed to a lower (or upper) triangular matrix, and for such matrices, the determinant equals the product of the entries on the main diagonal; this provides a method to calculate the determinant of any matrix. Finally, the Laplace expansion expresses the determinant in terms of minors, that is, determinants of smaller matrices.Template:Sfnp This expansion can be used for a recursive definition of determinants (taking as starting case the determinant of a Template:Math matrix, which is its unique entry, or even the determinant of a Template:Math matrix, which is Template:Math), that can be seen to be equivalent to the Leibniz formula. Determinants can be used to solve linear systems using Cramer's rule, where the division of the determinants of two related square matrices equates to the value of each of the system's variables.Template:Sfnp

Eigenvalues and eigenvectors

Template:Main A number $λ$ and a nonzero vector Template:Math satisfying $𝐀 𝐯 = λ 𝐯$ are called an eigenvalue and an eigenvector of Template:Math, respectively.^[6]Template:Sfnp The number Template:Mvar is an eigenvalue of an Template:Math matrix Template:Math if and only if Template:Math is not invertible, which is equivalent toTemplate:Sfnp $det (𝐀 - λ 𝐈) = 0.$ The polynomial Template:Math in an indeterminate Template:Mvar given by evaluation of the determinant Template:Math is called the characteristic polynomial of Template:Math. It is a monic polynomial of degree Template:Mvar. Therefore the polynomial equation Template:Math has at most Template:Mvar different solutions, that is, eigenvalues of the matrix.Template:Sfnp They may be complex even if the entries of Template:Math are real.Template:Sfnp According to the Cayley–Hamilton theorem, Template:Math, that is, the result of substituting the matrix itself into its characteristic polynomial yields the zero matrix.Template:Sfnp

Computational aspects

Matrix calculations can be often performed with different techniques. Many problems can be solved by both direct algorithms and iterative approaches. For example, the eigenvectors of a square matrix can be obtained by finding a sequence of vectors Template:Math converging to an eigenvector when Template:Mvar tends to infinity.Template:Sfnp

To choose the most appropriate algorithm for each specific problem, it is important to determine both the effectiveness and precision of all the available algorithms. The domain studying these matters is called numerical linear algebra.Template:Sfnp As with other numerical situations, two main aspects are the complexity of algorithms and their numerical stability.

Determining the complexity of an algorithm means finding upper bounds or estimates of how many elementary operations such as additions and multiplications of scalars are necessary to perform some algorithm, for example, multiplication of matrices. Calculating the matrix product of two Template:Mvar-by-Template:Mvar matrices using the definition given above needs Template:Math multiplications, since for any of the Template:Math entries of the product, Template:Mvar multiplications are necessary. The Strassen algorithm outperforms this "naive" algorithm; it needs only Template:Math multiplications.Template:Sfnp Theoretically faster but impractical matrix multiplication algorithms have been developed,Template:Sfnp as have speedups to this problem using parallel algorithms or distributed computation systems such as MapReduce.Template:Sfnp

In many practical situations, additional information about the matrices involved is known. An important case concerns sparse matrices, that is, matrices whose entries are mostly zero. There are specifically adapted algorithms for, say, solving linear systems Template:Math for sparse matrices Template:Math, such as the conjugate gradient method.Template:Sfnp

An algorithm is, roughly speaking, numerically stable if little deviations in the input values do not lead to big deviations in the result. For example, one can calculate the inverse of a matrix by computing its adjugate matrix: $𝐀^{- 1} = adj (𝐀) / det (𝐀) .$ However, this 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, such as computing a matrix's inverse.Template:Sfnp

Decomposition

Template:Main There are several methods to render matrices into a more easily accessible form. They are generally referred to as matrix decomposition or matrix factorization techniques. These techniques are of interest because they can make computations easier.

The LU decomposition factors matrices as a product of lower (Template:Math) and an upper triangular matrices (Template:Math).Template:Sfnp 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.Template:Sfnp 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. Singular value decomposition (SVD) expresses any matrix Template:Math as a product Template:Math, where Template:Math and Template:Math are unitary matrices and Template:Math is a diagonal matrix.Template:Sfnp

An example of a matrix in Jordan normal form. The grey blocks are called Jordan blocks.

The eigendecomposition or diagonalization expresses Template:Math as a product Template:Math, where Template:Math is a diagonal matrix and Template:Math is a suitable invertible matrix.Template:Sfnp If Template:Math 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, that is to say matrices whose only nonzero entries are the eigenvalues Template:Math to Template:Mvar of Template:Math, placed on the main diagonal and possibly entries equal to one directly above the main diagonal, as shown at the right.Template:Sfnp Given the eigendecomposition, the Template:Mvarth power of Template:Math (that is, Template:Mvar-fold iterated matrix multiplication) can be calculated via $𝐀^{n} = ({𝐕 𝐃 𝐕}^{- 1})^{n} = {𝐕 𝐃 𝐕}^{- 1} {𝐕 𝐃 𝐕}^{- 1} \dots {𝐕 𝐃 𝐕}^{- 1} = {𝐕 𝐃}^{n} 𝐕^{- 1}$ 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 Template:Math instead. This can be used to compute the matrix exponential Template:Math, a need frequently arising in solving linear differential equations, matrix logarithms and square roots of matrices.Template:Sfnp To avoid numerically ill-conditioned situations, further algorithms such as the Schur decomposition can be employed.Template:Sfnp

Abstract algebraic aspects and generalizations

Matrices can be generalized in different ways. Abstract algebra uses matrices with entries in more general fields or even rings, while linear algebra codifies properties of matrices in the notion of linear maps. It is possible to consider matrices with infinitely many columns and rows. Another extension is tensors, which can be seen as higher-dimensional arrays of numbers, as opposed to vectors, which can often be realized as sequences of numbers, while matrices are rectangular or two-dimensional arrays of numbers.Template:Sfnp Matrices, subject to certain requirements tend to form groups known as matrix groups.Template:Sfnp Similarly under certain conditions matrices form rings known as matrix rings.Template:Sfnp Though the product of matrices is not in general commutative, certain matrices form fields sometimes called matrix fields.Template:Sfnp (However the term "matrix field" is ambiguous, also referring to certain forms of physical fields that continuously map points of some space to matrices.Template:Sfnp) In general, matrices over any ring and their multiplication can be represented as the arrows and composition of arrows in a category, the category of matrices over that ring. The objects of this category are natural numbers, representing the dimensions of the matrices.Template:Sfnp

Matrices with entries in a field or ring

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, that is, a set where addition, subtraction, multiplication, and division operations are defined and well-behaved, may be used instead of Template:Tmath or Template:Tmath, for example rational numbers or finite fields. For example, coding theory makes use of matrices over finite fields.Template:Sfnp Wherever eigenvalues are considered, as these are roots of a polynomial, they may exist only in a larger field than that of the entries of the matrix. For instance, they may be complex in the case of a matrix with real entries. The possibility to reinterpret the entries of a matrix as elements of a larger field (for example, to view a real matrix as a complex matrix whose entries happen to be all real) then allows considering each square matrix to possess a full set of eigenvalues.Template:Sfnp Alternatively one can consider only matrices with entries in an algebraically closed field, such as Template:Tmath from the outset.Template:Sfnp

Matrices whose entries are polynomials,Template:Sfnp and more generally, matrices with entries in a ring Template:Mvar are widely used in mathematics.Template:Sfnp Rings are a more general notion than fields in that a division operation need not exist. The very same addition and multiplication operations of matrices extend to this setting, too. The set Template:Math (also denoted Template:Math Template:Sfnp) of all square Template:Mvar-by-Template:Mvar matrices over Template:Mvar is a ring called matrix ring, isomorphic to the endomorphism ring of the left Template:Mvar-module Template:Mvar.Template:Sfnp If the ring Template:Mvar is commutative, that is, its multiplication is commutative, then the ring Template:Math is also an associative algebra over Template:Math. The determinant of square matrices over a commutative ring Template:Mvar can still be defined using the Leibniz formula; such a matrix is invertible if and only if its determinant is invertible in Template:Mvar, generalizing the situation over a field Template:Mvar, where every nonzero element is invertible.Template:Sfnp Matrices over superrings are called supermatrices.Template:Sfnp

Matrices do not always have all their entries in the same ringTemplate:Nbsp– or even in any ring at all. One special but common case is block matrices, which may be considered as matrices whose entries themselves are matrices. The entries need not be square matrices, and thus need not be members of any ring; but in order to multiply them, their sizes must fulfill certain conditions: each pair of submatrices that are multiplied in forming the overall product must have compatible sizes.Template:Sfnp

Relationship to linear maps

Linear maps $ℝ^{n} \to ℝ^{m}$ are equivalent to Template:Mvar-by-Template:Mvar matrices, as described above. More generally, any linear map Template:Math between finite-dimensional vector spaces can be described by a matrix Template:Math, after choosing bases Template:Math of Template:Mvar, and Template:Math of Template:Mvar (so Template:Mvar is the dimension of Template:Mvar and Template:Mvar is the dimension of Template:Mvar), which is such that $f (𝐯_{j}) = \sum_{i = 1}^{m} a_{i, j} 𝐰_{i} for j = 1, \dots, n .$ In other words, column Template:Mvar of Template:Math expresses the image of Template:Math in terms of the basis vectors Template:Math of Template:Mvar; thus this relation uniquely determines the entries of the matrix Template:Math. The matrix depends on the choice of the bases: different choices of bases give rise to different, but equivalent matrices.Template:Sfnp Many of the above concrete notions can be reinterpreted in this light, for example, the transpose matrix Template:Math describes the transpose of the linear map given by Template:Math, concerning the dual bases.Template:Sfnp

These properties can be restated more naturally: the category of matrices with entries in a field $k$ with multiplication as composition is equivalent to the category of finite-dimensional vector spaces and linear maps over this field.Template:Sfnp

More generally, the set of Template:Math matrices can be used to represent the Template:Mvar-linear maps between the free modules Template:Mvar and Template:Mvar for an arbitrary ring Template:Mvar with unity. When Template:Math composition of these maps is possible, and this gives rise to the matrix ring of Template:Math matrices representing the endomorphism ring of Template:Mvar.Template:Sfnp

Matrix groups

Template:Main A group is a mathematical structure consisting of a set of objects together with a binary operation, that is, an operation combining any two objects to a third, subject to certain requirements.Template:Sfnp A group in which the objects are invertible Template:Tmath matrices and the group operation is matrix multiplication is called a matrix group of degree Template:Tmath.Template:Sfnp Every such matrix group is a subgroup of (that is, a smaller group contained within) the group of all invertible Template:Tmath matrices, the general linear group of degree Template:Tmath.Template:Sfnp

Any property of square matrices that is preserved under matrix products and inverses can be used to define a matrix group. For example, the set of all Template:Tmath matrices whose determinant is Template:Math form a group called the special linear group of degree Template:Tmath.Template:Sfnp The set of orthogonal matrices, determined by the condition $𝐌^{T} 𝐌 = 𝐈,$ form the orthogonal group.Template:Sfnp Every orthogonal matrix has determinant Template:Math or Template:Math. Orthogonal matrices with determinant Template:Math form a group called the special orthogonal group.Template:Sfnp

Every finite group is isomorphic to a matrix group, as one can see by considering the regular representation of the symmetric group.Template:Sfnp General groups can be studied using matrix groups, which are comparatively well understood, using representation theory.^[7]

Infinite matrices

It is also possible to consider matrices with infinitely many rows and columns.^[8] The basic operations introduced above are defined 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.Template:Sfnp An easy way to circumvent this issue is to restrict to finitary matrices all of whose rows (or columns) contain only finitely many nonzero terms.Template:Sfnp As in the finite case (see above), where matrices describe linear maps, infinite matrices can be used to describe operators on Hilbert spaces, where convergence and continuity questions arise. However, the explicit point of view of matrices tends to obfuscate the matter,^[9] 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.

Empty matrix

An empty matrix is a matrix in which the number of rows or columns (or both) is zero.^[10]^[1] Empty matrices can be a useful base case for certain recursive constructions,Template:Sfnp and can help to deal with maps involving the zero vector space.Template:Sfnp For example, if Template:Math is a Template:Math matrix and Template:Math is a Template:Math matrix, then Template:Math is the Template:Math zero matrix corresponding to the null map from a 3-dimensional space Template:Mvar to itself, while Template:Math is a Template:Math matrix. There is no common notation for empty matrices, but most computer algebra systems allow creating and computing with them.^[11] The determinant of the Template:Math matrix is conventionally defined to be 1, consistent with the empty product occurring in the Leibniz formula for the determinant.Template:Sfnp This value is also needed for consistency with the Template:Math case of the Desnanot–Jacobi identity relating determinants to the determinants of smaller matrices.Template:Sfnp

Matrices with entries in a semiring

A semiring is similar to a ring, but elements need not have additive inverses, therefore one cannot do subtraction freely there. The definition of addition and multiplication of matrices with entries in a ring applies to matrices with entries in a semiring without modification. Matrices of fixed size with entries in a semiring form a commutative monoid $Mat (m, n; R)$ under addition.Template:Sfnp Square matrices of fixed size with entries in a semiring form a semiring $Mat (n; R)$ under addition and multiplication.Template:Sfnp

The determinant of an Template:Math square matrix $M$ with entries in a commutative semiring $R$ cannot be defined in general because the definition would involve additive inverses of semiring elements. What plays its role instead is the pair of positive and negative determinants

\underset{+}{det} M = \sum_{σ \in Alt (n)} M_{1 σ (1)} \dots M_{n σ (n)}

\underline{det} M = \sum_{σ \in Sym (n) ∖ Alt (n)} M_{1 σ (1)} \dots M_{n σ (n)}

where the sums are taken over even permutations and odd permutations, respectively.Template:Sfnp Template:Sfnp

Matrices with entries in a category

Matrices and their multiplication can be defined with entries objects of a category equipped with a "tensor product" similar to multiplication in a ring, having coproducts similar to addition in a ring, in that the former is distributive over the latter.Template:Sfnp However, the multiplication thus defined may be only associative in a sense weaker than usual. These are part of a bigger structure called the bicategory of matrices. The complete description of the above summary for interested readers follows.

Let $(𝒞, \otimes, I)$ be a monoidal category satisfying the following two conditions:

All (small) coproducts exist; in particular, let $\emptyset$ be an initial object.
The functor $\otimes$ is distributive over coproducts; i.e., for all object $X$ and a family of objects $(Y_{i})_{i \in I}$ in $𝒞$ , the canonical $𝒞$ -morphisms $\underset{i \in I}{∐} (X \otimes Y_{i}) \to X \otimes \underset{i \in I}{∐} Y_{i}$ $\underset{i \in I}{∐} (Y_{i} \otimes X) \to (\underset{i \in I}{∐} Y_{i}) \otimes X$ are isomorphisms. In particular, the canonical morphisms $\emptyset \to X \otimes \emptyset$ and $\emptyset \to \emptyset \otimes X$ are isomorphisms.

Then, the bicategory of $𝒞$ -matrices $Mat (𝒞)$ is as follows:Template:Sfnp

The objects are the sets.
A 1-morphism M:A→B is a map M:A×B→Ob⁡(𝒞); this is just a matrix over 𝒞.
- The composition of 1-morphisms $M : A \to B$ and $N : B \to C$ , which can be understood as matrix multiplication, is $(N \circ M) (a, c) = \underset{b \in B}{∐} M (a, b) \otimes N (b, c) .$
- The identity 1-morphism on $A$ is ${id}_{A} (a, b) = {\begin{matrix} I & a = b \\ \emptyset & a \neq b \end{matrix} .$
A 2-morphism between 1-morphisms $M, N : A \to B$ is a family of $𝒞$ -morphisms $(f_{a b} : M (a, b) \to N (a, b))_{(a, b) \in A \times B}$ . The definition of vertical and horizontal composition of 2-morphisms is natural: the vertical composition is componentwise composition of $𝒞$ -morphisms; the horizontal composition is that derived from the functoriality of $\otimes$ and the universal property of coproducts.

In general, the bicategory of matrices need not be a strict 2-category. For example, the composition of 1-morphisms may not be associative in the usual strict sense, but only up to coherent isomorphism.

Applications

There are numerous 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, text mining and automated thesaurus compilation makes use of document-term matrices such as tf-idf to track frequencies of certain words in several documents.Template:Sfnp

Complex numbers can be represented by particular real 2-by-2 matrices via $a + i b \leftrightarrow [\begin{matrix} a & - b \\ b & a \end{matrix}],$ under which 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 Template:Sfnp and Clifford algebras in general.Template:Sfnp

In game theory and economics, the payoff matrix encodes the payoff for two players, depending on which out of a given (finite) set of strategies the players choose.Template:Sfnp The expected outcome of the game, when both players play mixed strategies, is obtained by multiplying this matrix on both sides by vectors representing the strategies.Template:Sfnp The minimax theorem central to game theory is closely related to the duality theory of linear programs, which are often formulated in terms of matrix-vector products.Template:Sfnp

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.Template:Sfnp Computer graphics uses matrices to represent objects; to calculate transformations of objects using affine rotation matrices to accomplish tasks such as projecting a three-dimensional object onto a two-dimensional screen, corresponding to a theoretical camera observation; and to apply image convolutions such as sharpening, blurring, edge detection, and more.Template:Sfnp Matrices over a polynomial ring are important in the study of control theory.Template:Sfnp

Chemistry makes use of matrices in various ways, particularly since the use of quantum theory to discuss molecular bonding and spectroscopy. Examples are the overlap matrix and the Fock matrix used in solving the Roothaan equations to obtain the molecular orbitals of the Hartree–Fock method.Template:Sfnp

Graph theory

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

The adjacency matrix of a finite graph is a basic notion of graph theory.Template:Sfnp It records which vertices of the graph are connected by an edge. Matrices containing just two different values (Template:Math and Template:Math meaning for example "yes" and "no", respectively) are called logical matrices. The distance (or cost) matrix contains information about the distances of the edges.Template:Sfnp These concepts can be applied to websites connected by hyperlinks,Template:Sfnp or cities connected by roads etc., in which case (unless the connection network is extremely dense) the matrices tend to be sparse, that is, contain few nonzero entries. Therefore, specifically tailored matrix algorithms can be used in network theory.Template:Sfnp

Analysis and geometry

The Hessian matrix of a differentiable function $f : ℝ^{n} \to ℝ$ consists of the second derivatives of Template:Mvar concerning the several coordinate directions, that is,Template:Sfnp $H (f) = [\frac{\partial^{2} f}{\partial x_{i} \partial x_{j}}] .$

It encodes information about the local growth behavior of the function: given a critical point Template:Math, that is, a point where the first partial derivatives $\partial f / \partial x_{i}$ of Template:Mvar vanish, the function has a local minimum if the Hessian matrix is positive definite. Quadratic programming can be used to find global minima or maxima of quadratic functions closely related to the ones attached to matrices (see above).Template:Sfnp

Another matrix frequently used in geometrical situations is the Jacobi matrix of a differentiable map Template:Tmath. If Template:Math denote the components of Template:Mvar, then the Jacobi matrix is defined asTemplate:Sfnp $J (f) = {[\frac{\partial f_{i}}{\partial x_{j}}]}_{1 \leq i \leq m, 1 \leq j \leq n} .$ If Template:Math, and if the rank of the Jacobi matrix attains its maximal value Template:Mvar, Template:Mvar is locally invertible at that point, by the implicit function theorem.^[12]

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 a decisive influence on the set of possible solutions of the equation in question.Template:Sfnp

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 concerning a sufficiently fine grid, which in turn can be recast as a matrix equation.^[13]

Template:Clear

Probability theory and statistics

Two different Markov chains. The chart depicts the number of particles (of a total of 1000) in state "2". Both limiting values can be determined from the transition matrices, which are given by $[\begin{matrix} 0.7 & 0 \\ 0.3 & 1 \end{matrix}]$ (red) and $[\begin{matrix} 0.7 & 0.2 \\ 0.3 & 0.8 \end{matrix}]$ (black).

Stochastic matrices are square matrices whose rows are probability vectors, that is, whose entries are non-negative and sum up to one. Stochastic matrices are used to define Markov chains with finitely many states.Template:Sfnp A row of the stochastic matrix gives the probability distribution for the next position of some particle currently in the state that corresponds to the row. Properties of the Markov chain—like absorbing states, that is, states that any particle attains eventually—can be read off the eigenvectors of the transition matrices.Template:Sfnp

Statistics also makes use of matrices in many different forms.^[14] Descriptive statistics is concerned with describing data sets, which can often be represented as data matrices, which may then be subjected to dimensionality reduction techniques. The covariance matrix encodes the mutual variance of several random variables.Template:Sfnp Another technique using matrices are linear least squares, a method that approximates a finite set of pairs Template:Math, by a linear function $y_{i} \approx a x_{i} + b, i = 1, \dots, N$ which can be formulated in terms of matrices, related to the singular value decomposition of matrices.Template:Sfnp

Random matrices are matrices whose entries are random numbers, subject to suitable probability distributions, such as matrix normal distribution. Beyond probability theory, they are applied in domains ranging from number theory to physics.^[15]^[16]

Quantum mechanics and particle physics

The first model of quantum mechanics (Heisenberg, 1925) used infinite-dimensional matrices to define the operators that took over the role of variables like position, momentum and energy from classical physics.Template:Sfnp (This is sometimes referred to as matrix mechanics.Template:Sfnp) Matrices, both finite and infinite-dimensional, have since been employed for many purposes in quantum mechanics. One particular example is the density matrix, a tool used in calculating the probabilities of the outcomes of measurements performed on physical systems.Template:Sfnp Template:Sfnp

Linear transformations 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.Template:Sfnp 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 modern 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.Template:Sfnp

Another matrix serves as a key tool for describing the scattering experiments that 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.Template:Sfnp

Normal modes

A general application of matrices in physics is 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 the internal dynamics of molecules: the internal vibrations of systems consisting of mutually bound component atoms.Template:Sfnp They are also needed for describing mechanical vibrations, and oscillations in electrical circuits.Template:Sfnp

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 analysis: 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. There are two kinds of matrices, viz. a refraction matrix describing the refraction at a lens surface, and a translation matrix, describing the translation of the plane of reference to the next refracting surface, where another refraction matrix applies. The optical system, consisting of a combination of lenses and reflective elements, is simply described by the matrix resulting from the product of the components' matrices.Template:Sfnp

The Jones calculus models the polarization of a light source as a $2 \times 2$ vector, and the effects of optical filters on this polarization vector as a matrix.Template:Sfnp

Electronics

Electronic circuits that are composed of linear components (such as resistors, inductors and capacitors) obey Kirchhoff's circuit laws, which leads to a system of linear equations, which can be described with a matrix equation that relates the source currents and voltages to the resultant currents and voltages at each point in the circuit, and where the matrix entries are determined by the circuit.Template:Sfnp

History

Matrices have a long history of application in solving linear equations but they were known as arrays until the 1800s. The Chinese text The Nine Chapters on the Mathematical Art written in the 10th–2nd century BCE is the first example of the use of array methods to solve simultaneous equations^[17]. In 1545 Italian mathematician Gerolamo Cardano introduced the method to Europe when he published Ars Magna.Template:Sfnp The Japanese mathematician Seki used the same array methods to solve simultaneous equations in 1683.^[18] The Dutch mathematician Jan de Witt represented transformations using arrays in his 1659 book Elements of Curves (1659).Template:Sfnp Between 1700 and 1710 Gottfried Wilhelm Leibniz publicized the use of arrays for recording information or solutions and experimented with over 50 different systems of arrays.Template:Sfnp Cramer presented his rule in 1750.Template:Sfnp Template:Sfnp

This use of the term matrix in mathematics (an English word for "womb" in the 19th century, from Latin, as well as a jargon word in printing, in biology and in geology^[19]) was coined by James Joseph Sylvester in 1850,^[20] who understood a matrix as an object giving rise to several determinants today called minors, that is to say, determinants of smaller matrices that derive from the original one by removing columns and rows. In an 1851 paper, Sylvester explains:Template:Sfnp

Template:Blockquote

Arthur Cayley published a treatise on geometric transformations using matrices that were not rotated versions of the coefficients being investigated as had previously been done. Instead, he defined operations such as addition, subtraction, multiplication, and division as transformations of those matrices and showed the associative and distributive properties held. Cayley investigated and demonstrated the non-commutative property of matrix multiplication as well as the commutative property of matrix addition.Template:Sfnp Early matrix theory had limited the use of arrays almost exclusively to determinants and Cayley's abstract matrix operations were revolutionary. He was instrumental in proposing a matrix concept independent of equation systems. In 1858, Cayley published his A memoir on the theory of matricesTemplate:Sfnp Template:Sfnp in which he proposed and demonstrated the Cayley–Hamilton theorem.Template:Sfnp

The English mathematician Cuthbert Edmund Cullis was the first to use modern bracket notation for matrices in 1913 and he simultaneously demonstrated the first significant use of the notation Template:Math to represent a matrix where Template:Math refers to the Template:Mvarth row and the Template:Mvarth column.Template:Sfnp

The modern study of determinants sprang from several sources.Template:Sfnp Number-theoretical problems led Gauss to relate coefficients of quadratic forms, that is, expressions such as Template:Math, and linear maps in three dimensions to matrices. Eisenstein further developed these notions, including the remark that, in modern parlance, matrix products are non-commutative. Cauchy was the first to prove general statements about determinants, using as the definition of the determinant of a matrix Template:Math the following: replace the powers Template:Math by Template:Math in the polynomial $a_{1} a_{2} \dots a_{n} \prod_{i < j} (a_{j} - a_{i}),$ where $\prod$ denotes the product of the indicated terms. He also showed, in 1829, that the eigenvalues of symmetric matrices are real.Template:Sfnp Jacobi studied "functional determinants"—later called Jacobi determinants by Sylvester—which can be used to describe geometric transformations at a local (or infinitesimal) level, see above. Kronecker's Vorlesungen über die Theorie der Determinanten^[21] and Weierstrass's Zur Determinantentheorie,^[22] both published in 1903, first treated determinants axiomatically, as opposed to previous more concrete approaches such as the mentioned formula of Cauchy. At that point, determinants were firmly established.Template:Sfnp Template:Sfnp

Many theorems were first established for small matrices only, for example, the Cayley–Hamilton theorem was proved for Template:Math matrices by Cayley in the aforementioned memoir, and by Hamilton for Template:Math matrices. Frobenius, working on bilinear forms, generalized the theorem to all dimensions (1898). Also at the end of the 19th century, the Gauss–Jordan elimination (generalizing a special case now known as Gauss elimination) was established by Wilhelm Jordan. In the early 20th century, matrices attained a central role in linear algebra,Template:Sfnp partially due to their use in the classification of the hypercomplex number systems of the previous century.Template:Sfnp

The inception of matrix mechanics by Heisenberg, Born and Jordan led to studying matrices with infinitely many rows and columns.Template:Sfnp Later, von Neumann carried out the mathematical formulation of quantum mechanics, by further developing functional analytic notions such as linear operators on Hilbert spaces, which, very roughly speaking, correspond to Euclidean space, but with an infinity of independent directions.Template:Sfnp

Other historical usages of the word "matrix" in mathematics

The word has been used in unusual ways by at least two authors of historical importance.

Bertrand Russell and Alfred North Whitehead in their Principia Mathematica (1910–1913) use the word "matrix" in the context of their axiom of reducibility. They proposed this axiom as a means to reduce any function to one of lower type, successively, so that at the "bottom" (0 order) the function is identical to its extension:^[23]

Template:Blockquote

For example, a function Template:Math of two variables Template:Mvar and Template:Mvar can be reduced to a collection of functions of a single variable, such as Template:Mvar, by "considering" the function for all possible values of "individuals" Template:Mvar substituted in place of a variable Template:Mvar. And then the resulting collection of functions of the single variable Template:Mvar, that is, Template:Math, can be reduced to a "matrix" of values by "considering" the function for all possible values of "individuals" Template:Math substituted in place of variable Template:Mvar: $\forall b_{j} \forall a_{i} : ϕ (a_{i}, b_{j}) .$

Alfred Tarski in his 1941 Introduction to Logic used the word "matrix" synonymously with the notion of truth table as used in mathematical logic.Template:Sfnp

Notes

Template:Reflist Template:Reflist

References

External links

Template:Sister project links

Template:Linear algebra Template:Tensors Template:Matrix classes

Template:Authority control

↑ ^1.0 ^1.1 "A matrix having at least one dimension equal to zero is called an empty matrix", MATLAB Data Structures Template:Webarchive
↑ For example, for Template:Tmath, see Template:Harvtxt, p. 48; for Template:Tmath, see Template:Harvtxt, p. 50.
↑ Template:Citation.
↑ Template:Harvtxt. Note however that Greub follows a transposed convention of representing a transformation by multiplying a row vector by a matrix, rather than multiplying a matrix by a column vector, leading to the reversed order for the two matrices in the product that represents a composition.
↑ Template:Cite web
↑ Eigen means "own" in German and in Dutch. See Wiktionary.
↑ See any reference in representation theory or group representation.
↑ See the item "Matrix" in Template:Harvard citations.
↑ "Not much of matrix theory carries over to infinite-dimensional spaces, and what does is not so useful, but it sometimes helps." Template:Harvard citations.
↑ "Empty Matrix: A matrix is empty if either its row or column dimension is zero", Glossary Template:Webarchive, O-Matrix v6 User Guide
↑ The notation of empty matrix is used differently from some sources like Template:Harvp use Template:Nowrap resembling the zero matrix; Template:Harvp use Template:Nowrap
↑ Template:Harvard citations. For a more advanced, and more general statement see Template:Harvard citations.
↑ Template:Harvard citations. See also stiffness method.
↑ Template:Citation
↑ Template:Harvard citations
↑ Template:Harvard citations
↑ Template:Harvard citations cited by Template:Harvard citations
↑ Template:Cite book
↑ Template:Citation
↑ The earliest published example is J. J. Sylvester (1850) "Additions to the articles in the September number of this journal, 'On a new class of theorems,' and on Pascal's theorem," The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 37: 363-370. From page 369: "For this purpose, we must commence, not with a square, but with an oblong arrangement of terms consisting, suppose, of m lines and n columns. This does not in itself represent a determinant, but is, as it were, a Matrix out of which we may form various systems of determinants ... "
↑ Template:Harvard citations
↑ Template:Harvard citations
↑ Whitehead, Alfred North; and Russell, Bertrand (1913) Principia Mathematica to *56, Cambridge at the University Press, Cambridge UK (republished 1962) cf page 162ff.

[empty-1] 1.0 ^1.1 "A matrix having at least one dimension equal to zero is called an empty matrix", MATLAB Data Structures Template:Webarchive

[2] For example, for Template:Tmath, see Template:Harvtxt, p. 48; for Template:Tmath, see Template:Harvtxt, p. 50.

[3] Template:Citation.

[4] Template:Harvtxt. Note however that Greub follows a transposed convention of representing a transformation by multiplying a row vector by a matrix, rather than multiplying a matrix by a column vector, leading to the reversed order for the two matrices in the product that represents a composition.

[:3-5] Template:Cite web

[6] Eigen means "own" in German and in Dutch. See Wiktionary.

[7] See any reference in representation theory or group representation.

[8] See the item "Matrix" in Template:Harvard citations.

[9] "Not much of matrix theory carries over to infinite-dimensional spaces, and what does is not so useful, but it sometimes helps." Template:Harvard citations.

[10] "Empty Matrix: A matrix is empty if either its row or column dimension is zero", Glossary Template:Webarchive, O-Matrix v6 User Guide

[11] The notation of empty matrix is used differently from some sources like Template:Harvp use Template:Nowrap resembling the zero matrix; Template:Harvp use Template:Nowrap

[12] Template:Harvard citations. For a more advanced, and more general statement see Template:Harvard citations.

[13] Template:Harvard citations. See also stiffness method.

[14] Template:Citation

[15] Template:Harvard citations

[16] Template:Harvard citations

[17] Template:Harvard citations cited by Template:Harvard citations

[18] Template:Cite book

[19] Template:Citation

[20] The earliest published example is J. J. Sylvester (1850) "Additions to the articles in the September number of this journal, 'On a new class of theorems,' and on Pascal's theorem," The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 37: 363-370. From page 369: "For this purpose, we must commence, not with a square, but with an oblong arrangement of terms consisting, suppose, of m lines and n columns. This does not in itself represent a determinant, but is, as it were, a Matrix out of which we may form various systems of determinants ... "

[21] Template:Harvard citations

[22] Template:Harvard citations

[23] Whitehead, Alfred North; and Russell, Bertrand (1913) Principia Mathematica to *56, Cambridge at the University Press, Cambridge UK (republished 1962) cf page 162ff.

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Matrix (mathematics)

Definition

Size

Notation

Basic operations

Addition

Scalar multiplication

Transpose

Matrix multiplication

Row operations

Submatrix

Linear equations

Linear transformations

Square matrix

Main types

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Invertible matrix and its inverse

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Trace

Determinant

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Abstract algebraic aspects and generalizations

Matrices with entries in a field or ring

Relationship to linear maps

Matrix groups

Infinite matrices

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Graph theory

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Quantum mechanics and particle physics

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Other historical usages of the word "matrix" in mathematics

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