NationMaster - Encyclopedia: Matrix (mathematics)

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 can be added, multiplied, and decomposed in various ways, making them a key concept in linear algebra and matrix theory. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... For other uses, see Number (disambiguation). ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... In mathematics and linear algebra, a system of linear equations is a set of linear equations such as A standard problem is to decide if any assignment of values for the unknowns can satisfy all three equations simultaneously, and to find such an assignment if it exists. ... For other senses of this word, see coefficient (disambiguation). ... In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. ... Matrix theory is a branch of mathematics which focuses on the study of matrices. ...

In this article, the entries of a matrix are real or complex numbers unless otherwise noted. In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ...

Organization of a matrix

Image File history File links Matrix. ... Image File history File links Matrix. ...

Definitions and notations

The horizontal lines in a matrix are called rows and the vertical lines are called columns. A matrix with m rows and n columns is called an m-by-n matrix (written m × n) and m and n are called its dimensions. The dimensions of a matrix are always given with the number of rows first, then the number of columns. It is commonly said that an m-by-n matrix has an order of m × n ("order" meaning size). Two matrices of the same order whose corresponding entries are equivalent are considered equal.

Almost always capital letters denote matrices with the corresponding lower-case letters with two indices representing the entries. For example, the entry of a matrix A that lies in the i-th row and the j-th column is written as a_i,j and called the i,j entry or (i,j)-th entry of A. Alternative notations for that entry are A[i,j] or A_i,j. The row is always noted first, then the column. In this example, A (with no subscripts) would symbolize the entire matrix. In addition to using uppercase letters as symbols representing 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, $underset{r times c}{A}$ for an r-by-c matrix.

We often write $mathbf{A}:=(a_{i,j})_{i=1,ldots,m;,,j=1,ldots,n}$ or $mathbf{A}:=(a_{i,j})_{m times n}$ 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. In some programming languages, the numbering of rows and columns starts at zero. Texts which use any such language extensively, frequently follow that convention, so we have 0 ≤ i ≤ m-1 and 0 ≤ j ≤ n-1.

A matrix where one of the dimensions equals one is often called a vector, and 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. In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... In linear algebra, a column vector is an m Ã— 1 matrix, i. ... In linear algebra, a row vector is a 1 Ã— n matrix, that is, a matrix consisting of a single row: The transpose of a row vector is a column vector. ...

Mathematical definition

An $,m times n,,(m, n in mathbb{N})$ matrix $mathbf{A},$ is a function $mathbf{A}colon {1, 2, ldots, m} times {1, 2, ldots, n} to mathbf{S},,,$ where $mathbf{S},$ is any non-empty set. Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... The empty set is the set containing no elements. ...

$({1, 2, ldots, m} times {1, 2, ldots, n},$ is the Cartesian product of sets ${1, 2, ldots, m},$ and ${1, 2, ldots, n}.),$ In mathematics, the Cartesian product is a direct product of sets. ...

We say that matrix $mathbf{A}$ is a matrix over the set $mathbf{S}$ . Important thing to note is that, if we want to have matrix algebra, the set $mathbf{S},$ must be a ring and matrix $mathbf{A}$ must be a square matrix (see Square matrices and related definitions below for further explanation). Since the set of all square matrices over a ring is also a ring, matrix algebra is usually called matrix ring. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... 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. ... In abstract algebra the matrix ring M(n,R) is set of all n-by-n matrices over an arbitrary ring R. This forms a ring under matrix addition and multiplication. ...

Since this article mainly considers matrices over real numbers, matrices shown here are actually functions $mathbf{A}colon {1, 2, ldots, m} times {1, 2, ldots, n} to mathbb{R}.,,$ In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...

Example

The matrix

$mathbf{A} = begin{bmatrix} 9 & 8 & 6 1 & 2 & 7 4 & 9 & 2 6 & 0 & 5 end{bmatrix}$ or $mathbf{A} = begin{pmatrix} 9 & 8 & 6 1 & 2 & 7 4 & 9 & 2 6 & 0 & 5 end{pmatrix}$

is a $4times 3$ matrix. The element $a 2,3$ or $mathbf{A}[2,3]$ is 7. In terms of the mathematical definition given above, this matrix is a function $mathbf{A}colon {1, 2, 3, 4} times {1, 2, 3} to mathbb{R},$ and, for example, $mathbf{A}((2, 3)) = 7,$ and $mathbf{A}((3, 1)) = 4.,$

The matrix

$mathbf{R} = begin{bmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 end{bmatrix}$

is a $1times 9$ matrix, or 9-element row vector.

Adding and multiplying matrices

Sum

Main article: Matrix addition

Two or more matrices of identical dimensions m and n can be added. Given m-by-n matrices A and B, their sum A+B is the m-by-n matrix computed by adding corresponding elements: The operations on matrices differ from similar operations of scalar algebra in several respects. ...

$begin{align} mathbf{A}+mathbf{B} &= (a_{i,j})_{1le i le m;, 1le j le n} + (b_{i,j})_{1le i le m;, 1le j le n} &= (a_{i,j}+b_{i,j})_{1le i le m; 1le j le n} end{align}$

For example:

$begin{bmatrix} 1 & 3 & 1 1 & 0 & 0 1 & 2 & 2 end{bmatrix} + begin{bmatrix} 0 & 0 & 5 7 & 5 & 0 2 & 1 & 1 end{bmatrix} = begin{bmatrix} 1+0 & 3+0 & 1+5 1+7 & 0+5 & 0+0 1+2 & 2+1 & 2+1 end{bmatrix} = begin{bmatrix} 1 & 3 & 6 8 & 5 & 0 3 & 3 & 3 end{bmatrix}$

Another, much less often used notion of matrix addition is the direct sum. The usual matrix addition is defined for two matrices of same dimensions. ...

Scalar multiplication

Main article: Scalar multiplication

Given a matrix A and a number c, the scalar multiplication cA is computed by multiplying every element of A by the scalar c (i.e. $(cmathbf{A})_{i,j} = c cdot a_{i,j}$ ). For example: In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). ... In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). ... In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector. ...

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

Matrix addition and scalar multiplication turn the set $text{M}(m,n,mathbb{R})$ of all $m$ -by- $n$ matrices with real entries into a real vector space of dimension $mcdot n$ . In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ...

Matrix multiplication

Main article: Matrix multiplication

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. The middle dot ( $cdot$ ) is not used to indicate matrix multiplication (it is used for scalar multiplication). 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 given by: This article gives an overview of the various ways to perform matrix multiplication. ...

$(mathbf{AB})_{i,j} = a_{i,1} b_{1,j} + a_{i,2} b_{2,j} + ldots + a_{i,n} b_{n,j}$

for each pair $(i, j)$ . For example:

$begin{bmatrix} 1 & 0 & 2 -1 & 3 & 1 end{bmatrix} times begin{bmatrix} 3 & 1 2 & 1 1 & 0 end{bmatrix} = begin{bmatrix} ( 1 times 3 + 0 times 2 + 2 times 1) & ( 1 times 1 + 0 times 1 + 2 times 0) (-1 times 3 + 3 times 2 + 1 times 1) & (-1 times 1 + 3 times 1 + 1 times 0) end{bmatrix}$

$= begin{bmatrix} 5 & 1 4 & 2 end{bmatrix}.$

Matrix multiplication has the following properties:

(AB)C = A(BC) for all k-by-m matrices A, m-by-n matrices B and n-by-p matrices C ("associativity").
(A+B)C = AC+BC for all m-by-n matrices A and B and n-by-k matrices C ("right distributivity").
C(A+B) = CA+CB for all m-by-n matrices A and B and k-by-m matrices C ("left distributivity").

Matrix multiplication is not commutative; that is, given matrices A and B and their product defined, then generally AB $ne$ BA. It may also happen that AB is defined but BA is not defined. Example showing the commutativity of addition (3 + 2 = 2 + 3) For other uses, see Commute (disambiguation). ...

Besides the ordinary matrix multiplication just described, there exist other operations on matrices that can be considered forms of multiplication, such as the Hadamard product and the Kronecker product. In mathematics, the Kronecker product, denoted by , is an operation on two matrices of arbitrary size resulting in a block matrix. ...

Linear transformations, ranks and transpose

Main article: Transformation matrix

Main article: Transpose

Matrices can conveniently represent linear transformations because matrix multiplication neatly corresponds to the composition of maps, as will be described next. This same property makes them powerful data structures in high-level programming languages. In linear algebra, linear transformations can be represented by matrices. ... In linear algebra, the transpose of a matrix A is another matrix AT (also written Atr, tA, or Aâ€²) created by any one of the following equivalent actions: write the rows of A as the columns of AT write the columns of A as the rows of AT reflect A... In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...

Here and in the sequel we identify Rⁿ with the set of "columns" or n-by-1 matrices. For every linear map f : Rⁿ → R^m there exists a unique m-by-n matrix A such that f(x) = Ax for all x in Rⁿ. We say that the matrix A "represents" the linear map f. Now if the k-by-m matrix B represents another linear map g : R^m → R^k, then the linear map g o f is represented by BA. This follows from the above-mentioned associativity of matrix multiplication.

More generally, a linear map from an n-dimensional vector space to an m-dimensional vector space is represented by an m-by-n matrix, provided that bases have been chosen for each. In linear algebra, a basis is a set of vectors that, in a linear combination, can represent every vector in a given vector space, and such that no element of the set can be represented as a linear combination of the others. ...

The rank of a matrix A is the dimension of the image of the linear map represented by A; this is the same as the dimension of the space generated by the rows of A, and also the same as the dimension of the space generated by the columns of A. In linear algebra, the column rank (row rank respectively) of a matrix A with entries in some field is defined to be the maximal number of columns (rows respectively) of A which are linearly independent. ... In mathematics, the dimension of a vector space V is the cardinality (i. ... In mathematics, the image of an element x in a set X under the function f : X â†’ Y, denoted by f(x), is the unique y in Y that is associated with x. ...

The transpose of an m-by-n matrix A is the n-by-m matrix A^tr (also sometimes written as A^T or ^tA) formed by turning rows into columns and columns into rows, i.e. A^tr[i, j] = A[j, i] for all indices i and j. If A describes a linear map with respect to two bases, then the matrix A^tr describes the transpose of the linear map with respect to the dual bases, see dual space. In linear algebra, the transpose of a matrix A is another matrix AT (also written Atr, tA, or Aâ€²) created by any one of the following equivalent actions: write the rows of A as the columns of AT write the columns of A as the rows of AT reflect A... In mathematics, any vector space V has a corresponding dual vector space (or just dual space for short) consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra. ...

We have (A + B)^tr = A^tr + B^tr and (AB)^tr = B^tr A^tr.

Square matrices and related definitions

A square matrix is a matrix which has the same number of rows and columns. The set of all square n-by-n matrices, together with matrix addition and matrix multiplication is a ring. Unless n = 1, this ring is not commutative. In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ...

M(n, R), the ring of real square matrices, is a real unitary associative algebra. M(n, C), the ring of complex square matrices, is a complex associative algebra. In mathematics, an associative algebra is a vector space (or more generally, a module) which also allows the multiplication of vectors in a distributive and associative manner. ...

The unit matrix or identity matrix I_n, with elements on the main diagonal set to 1 and all other elements set to 0, satisfies MI_n = M and I_nN = N for any m-by-n matrix M and n-by-k matrix N. For example, if n = 3: In linear algebra, the identity matrix of size n is the n-by-n square matrix with ones on the main diagonal and zeros elsewhere. ... In linear algebra, the main diagonal of a square matrix is the diagonal which runs from the top left corner to the bottom right corner. ...

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

The identity matrix is the identity element in the ring of square matrices.

Invertible elements in this ring are called invertible matrices or non-singular matrices. An n by n matrix A is invertible if and only if there exists a matrix B such that In linear algebra, an n-by-n (square) matrix A is called invertible or non-singular if there exists an n-by-n matrix B such that where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ...

AB = I_n ( = BA).

In this case, B is the inverse matrix of A, denoted by A⁻¹. The set of all invertible n-by-n matrices forms a group (specifically a Lie group) under matrix multiplication, the general linear group. In mathematics and especially linear algebra, an n-by-n matrix A is called invertible, non-singular or regular if there exists another n-by-n matrix B such that AB = BA = In, where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ... This picture illustrates how the hours on a clock form a group under modular addition. ... In mathematics, a Lie group, named after Norwegian mathematician Sophus Lie (IPA pronunciation: , sounds like Lee), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. ... In mathematics, the general linear group of degree n is the set of nÃ—n invertible matrices, together with the operation of ordinary matrix multiplication. ...

If λ is a number and v is a non-zero vector such that Av = λv, then we call v an eigenvector of A and λ the associated eigenvalue. (Eigen means "own" in German and in Dutch.) The number λ is an eigenvalue of A if and only if A−λI_n is not invertible, which happens if and only if p_A(λ) = 0. Here p_A(x) is the characteristic polynomial of A. This is a polynomial of degree n and has therefore n complex roots (counting multiple roots according to their multiplicity). In this sense, every square matrix has n complex eigenvalues. In linear algebra, the eigenvectors (from the German eigen meaning own) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ... In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ... In linear algebra, one associates a polynomial to every square matrix, its characteristic polynomial. ... In mathematics, a polynomial is an expression that is constructed from one variable or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ...

The determinant of a square matrix A is the product of its n eigenvalues, but it can also be defined by the Leibniz formula. Invertible matrices are precisely those matrices with nonzero determinant. In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every nÃ—n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ... In algebra, the Leibniz formula expresses the determinant of a square matrix in terms of permutations of the matrix elements. ...

The Gaussian elimination algorithm is of central importance: it can be used to compute determinants, ranks and inverses of matrices and to solve systems of linear equations. In linear algebra, Gaussian elimination is an algorithm that can be used to determine the solutions of a system of linear equations, to find the rank of a matrix, and to calculate the inverse of an invertible square matrix. ... In mathematics and linear algebra, a system of linear equations is a set of linear equations such as A standard problem is to decide if any assignment of values for the unknowns can satisfy all three equations simultaneously, and to find such an assignment if it exists. ...

The trace of a square matrix is the sum of its diagonal entries, which equals the sum of its n eigenvalues. In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i. ... For the square matrix section, see square matrix. ...

Matrix exponential is defined for square matrices, using power series. In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. ... In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ...

Special types of matrices

In many areas in mathematics, matrices with certain structure arise. A few important examples are

Symmetric matrices are such that elements symmetric about the main diagonal (from the upper left to the lower right) are equal, that is, $a_{i,j}=a_{j,i} Leftrightarrow mathbf{A}^mathrm{T} = mathbf{A}$ .
Skew-symmetric matrices are such that elements symmetric about the main diagonal are the negative of each other, that is, $a_{i,j}=-a_{j,i} Leftrightarrow mathbf{A}^mathrm{T}=-mathbf{A}$ . In a skew-symmetric matrix, all diagonal elements are zero, that is, $a_{i,i}=-a_{i,i}Rightarrow a_{i,i}=0$ .
Hermitian (or self-adjoint) matrices are such that elements symmetric about the diagonal are each others complex conjugates, that is, $a_{i,j}=overline{a}_{j,i} Leftrightarrow mathbf{A}^mathrm{H} = mathbf{A}$ , where $overline{z}$ signifies the complex conjugate of a complex number $z$ and $,! mathbf{A}^mathrm{H}$ the conjugate transpose of A.
Toeplitz matrices have common elements on their diagonals, that is, $,! a_{i,j}=a_{i+1,j+1}$ .
Stochastic matrices are square matrices whose rows are probability vectors; they are used to define Markov chains.
A square matrix A is called idempotent if $mathbf{A}^2=mathbf{AA}=mathbf{A}$ .

For a more extensive list see list of matrices. In linear algebra, a symmetric matrix is a matrix that is its own transpose. ... In linear algebra, a skew-symmetric (or antisymmetric) matrix is a square matrix A whose transpose is also its negative; that is, it satisfies the equation: AT = −A or in component form, if A = (aij): aij = − aji for all i and j. ... A Hermitian matrix (or self-adjoint matrix) is a square matrix with complex entries which is equal to its own conjugate transpose â€” that is, the element in the ith row and jth column is equal to the complex conjugate of the element in the jth row and ith column, for... In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. ... In mathematics, the conjugate transpose, Hermitian transpose, or adjoint matrix of an m-by-n matrix A with complex entries is the n-by-m matrix A* obtained from A by taking the transpose and then taking the complex conjugate of each entry. ... In the mathematical discipline of linear algebra, a Toeplitz matrix or diagonal-constant matrix, named after Otto Toeplitz, is a matrix in which each descending diagonal from left to right is constant. ... In mathematics, especially in probability theory and statistics, and also in linear algebra and computer science, a left stochastic matrix is a square matrix whose columns are probability vectors, i. ... In mathematics and statistics, a probability vector or stochastic vector is a vector with non-negative entries that add up to one. ... In mathematics, a Markov chain, named after Andrey Markov, is a discrete-time stochastic process with the Markov property. ... In mathematics, an idempotent element is an element which, intuitively, leaves something unchanged. ... Listed below are some important classes of matrices used in mathematics: (0,1)-matrix or binary matrix - a matrix with all elements either 0 or 1. ...

Matrices in abstract algebra

If we start with a ring R, we can consider the set M(m,n, R) of all m by n matrices with entries in R. Addition and multiplication of these matrices can be defined as in the case of real or complex matrices (see above). 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. In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... In abstract algebra, one associates to certain objects a ring, the objects endomorphism ring, which encodes several internal properties of the object. ... In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalars to lie in a field, the scalars may lie in an arbitrary ring. ...

Similarly, if the entries are taken from a semiring S, matrix addition and multiplication can still be defined as usual. The set of all square n×n matrices over S is itself a semiring. Note that fast matrix multiplication algorithms such as the Strassen algorithm generally only apply to matrices over rings and will not work for matrices over semirings that are not rings. In abstract algebra, a semiring is an algebraic structure, similar to a ring, but without additive inverses. ... In the mathematical discipline of linear algebra, the Strassen algorithm, named after Volker Strassen, is an algorithm used for matrix multiplication. ...

If R is a commutative ring, then M(n, R) is a unitary associative algebra over R. It is then also meaningful to define the determinant of square matrices using the Leibniz formula; a matrix is invertible if and only if its determinant is invertible in R. In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... In mathematics, an associative algebra is a vector space (or more generally, a module) which also allows the multiplication of vectors in a distributive and associative manner. ... In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every nÃ—n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ... In algebra, the Leibniz formula expresses the determinant of a square matrix in terms of permutations of the matrix elements. ...

All statements mentioned in this article for real or complex matrices remain correct for matrices over an arbitrary field. In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...

Matrices over a polynomial ring are important in the study of control theory. In abstract algebra, a polynomial ring is the set of polynomials in one or more variables with coefficients in a ring. ... For control theory in psychology and sociology, see control theory (sociology). ...

History

The study of matrices is quite old. A 3-by-3 magic square appears in Chinese literature dating from as early as 650 BC.^[1] In recreational mathematics, a magic square of order n is an arrangement of nÂ² numbers, usually distinct integers, in a square, such that the n numbers in all rows, all columns, and both diagonals sum to the same constant. ... Chinese literature spans back thousands of years, from the earliest recorded dynastic court archives to the matured fictional novel arising in the medieval period to entertain the masses of literate Chinese. ...

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.^[2] 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. A linear equation in algebra is an equation which is constructed by equating two linear functions. ... The Nine Chapters on the Mathematical Art (ä¹ç« ç®—è¡“) is a Chinese mathematics book, probably composed in the 1st century AD, but perhaps as early as 200 BC. This book is the earliest surviving mathematical text from China that has come down to us by being copied by scribes and (centuries later... In mathematics and linear algebra, a system of linear equations is a set of linear equations such as A standard problem is to decide if any assignment of values for the unknowns can satisfy all three equations simultaneously, and to find such an assignment if it exists. ... In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every nÃ—n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ... Kowa Seki (Seki Takakazu, 関 孝和) (1642? – October 24, 1708) was a Japanese mathematician. ... Leibniz redirects here. ... Events January 11 - Eruption of Mt. ...

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.^[1] In the history of mathematics, Islamic mathematics or Arabic mathematics refers to the mathematics developed by the Islamic civilization between 622 and 1600. ... For other uses, see Arab (disambiguation). ... Map of South Asia (see note) This article deals with the geophysical region in Asia. ... This article is under construction. ... The astronomy and the astrology of Ancient India (Jyotisha) is based upon sidereal calculations. ... Combinatorics is a branch of mathematics that studies finite collections of objects that satisfy specified criteria, and is in particular concerned with counting the objects in those collections (enumerative combinatorics) and with deciding whether certain optimal objects exist (extremal combinatorics). ... Baghdad (Arabic: ) is the capital of Iraq and of Baghdad Governorate. ... Events Hugh Capet, a distant relative of the last Carolingian king of the Franks, is crowned King of France, beginning the Capetian dynasty and, arguably, modern French history. ... The Encyclopedia of the Brethren of Purity (also variously known as the Epistles of the Brethren of Sincerity, the Epistles of the Brethren of Purity or Epistles of the Brethren of Purity and Loyal Friends; Arabic: Rasail ikhwan as-safa wa khillan al-wafa ) was an encyclopedia written by...

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. Gabriel Cramer Gabriel Cramer (July 31, 1704 - January 4, 1752) was a Swiss mathematician, born in Geneva. ... Cramers rule is a theorem in linear algebra, which gives the solution of a system of linear equations in terms of determinants. ... Year 1750 (MDCCL) was a common year starting on Thursday (link will display the full calendar) of the Gregorian calendar (or a common year starting on Monday [1] of the 11-day slower Julian calendar). ... Johann Carl Friedrich Gauss or GauÃŸ ( ; Latin: ) (30 April 1777 â€“ 23 February 1855) was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, electrostatics, astronomy, and optics. ... Wilhelm Jordan (March 1, 1842 - April 17, 1899) was a German geodesist who did surveys in Germany and Africa and founded the German geodesy journal. ... In mathematics, Gaussian elimination or Gauss-Jordan elimination, named after Carl Friedrich Gauss and Wilhelm Jordan, is an algorithm in linear algebra for determining the solutions of a system of linear equations, for determining the rank of a matrix, and for calculating the inverse of an invertible square matrix. ...

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. Year 1848 (MDCCCXLVIII) was a leap year starting on Saturday (link will display the full calendar) of the Gregorian Calendar (or a leap year starting on Monday of the 12-day slower Julian calendar). ... James Joseph Sylvester James Joseph Sylvester (September 3, 1814 - March 15, 1897) was an English mathematician and lawyer. ... Arthur Cayley (August 16, 1821 - January 26, 1895) was a British mathematician. ... For other persons named William Hamilton, see William Hamilton (disambiguation). ... Hermann GÃ¼nther Grassmann (April 15, 1809, Stettin â€“ September 26, 1877, Stettin) was a German polymath, renowned in his day as a linguist and now admired as a mathematician. ... A picture of Frobenius Ferdinand Georg Frobenius (October 26, 1849 â€“ August 3, 1917) was a German mathematician, best-known for his contributions to the theory of differential equations and to group theory. ... For other persons named John Neumann, see John Neumann (disambiguation). ...

Olga Taussky-Todd (1906-1995) used matrix theory to investigate an aerodynamic phenomenon called fluttering or aeroelasticity during WWII. Olga Taussky Todd (August 30, 1906, Olomouc, then Austria-Hungary - October 7, 1995, Pasadena, California) was a mathematician. ... Flutter: In electronics, rapid variation of signal parameters, such as amplitude, phase, and frequency. ... Aeroelasticity is the science which studies the interaction among inertial, elastic, and aerodynamic forces. ... Combatants Allied powers: China France Great Britain Soviet Union United States and others Axis powers: Germany Italy Japan and others Commanders Chiang Kai-shek Charles de Gaulle Winston Churchill Joseph Stalin Franklin Roosevelt Adolf Hitler Benito Mussolini Hideki TÅjÅ Casualties Military dead: 17,000,000 Civilian dead: 33,000...

Education

Matrices were traditionally taught as part of linear algebra in college, or with calculus. With the adoption of integrated mathematics texts for use in high school in the 1990s, they have been included by many such texts such as the Core-Plus Mathematics Project which are often targeted as early as the ninth grade, or earlier for honors students. They often require the use of graphing calculators such as the TI-83 which can perform complex operations such as matrix inversion with a single keystroke. Integrated mathematics is a style of mathematics education which integrates many topics or strands of mathematics in a real-life context. ... The Core-Plus Mathematics Project is an NCTM-standards-based high school mathematics curriculum development project funded by the National Science Foundation. ... The TI-83 series graphing calculators are manufactured by Texas Instruments. ...

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 catridges to add BASIC commands for matrices. Some computer languages such as APL, were designed to manipulate matrices, and mathematical programs such as Mathematica, along with Maple and Matlab are also used to aid computing with matrices. The Hewlett-Packard HP 9830A was the top of the line of the 9800 series programmable calculators, introduced in 1972. ... APL (for A Programming Language) is an array programming language based on a notation invented in 1957 by Kenneth E. Iverson while at Harvard University. ... For other uses, see Mathematica (disambiguation). ... Maple is a general-purpose commercial mathematics software package. ... Not to be confused with Matlab Upazila in Chandpur District, Bangladesh. ...

Applications

Encryption

See also: Matrix encryption

Matrices can be used to encrypt numerical data. Encryption is done by multiplying the data matrix with a key matrix. Decryption is done simply by multiplying the encrypted matrix with the inverse of the key. In cryptography, matrix encryption is a relatively simple method of using matrices to encrypt text or data. ...

Computer graphics

See also: Transformation matrix

4×4 transformation matrices are commonly used in computer graphics. 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. In linear algebra, linear transformations can be represented by matrices. ...

References

^ ^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., Prentice-Hall, p. 1.

External links

Wikibooks Linear Algebra has a page on the topic of

Matrices

Resources
- Matrix name and history: very brief overview, ualr.edu
- Introduction to Matrix Algebra: definitions and properties, xycoon.com
- Matrix Algebra, sosmath.com
- The Matrix Reference Manual, Imperial College
- An online textbook on Introduction to Matrix Algebra at Holistic Numerical Methods Institute
- Applied examples of matrices used in graphical game programming, Riemer's DirectX Tutorials

Online Matrix Calculators
- easycalculation.com
- bluebit.gr
- wims.unice.fr
- Matrix Calculator at SolveMyMath - Calculate the Determinant, Trace, Transpose and Inverse Matrix of a Matrix

Freeware
- MATRIX 2.1 Excel add-in, foxes
- MacAnova, University of Minnesota School of Statistics

Image File history File links Wikibooks-logo-en. ... Wikibooks logo Wikibooks, previously called Wikimedia Free Textbook Project and Wikimedia-Textbooks, is a wiki for the creation of books. ... Royal School of Mines Entrance Imperial College London is a college of the University of London which focuses on science and technology, and is located in South Kensington in London. ... This article is about the oldest and largest campus of the University of Minnesota. ...

Categories: Abstract algebra | Linear algebra | Matrices

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