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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and A number is an Abstract object, tokens of which are Symbols used in Counting and measuring. In Abstract algebra, a semiring is an Algebraic structure similar to a ring, but without the requirement that each element must have an Additive inverse Matrices are used to describe linear equations, keep track of the coefficients of linear transformations and to record data that depend on multiple parameters. In Mathematics, a system of linear equations (or linear system) is a collection of Linear equations involving the same set of Variables For example In Mathematics, a coefficient is a Constant multiplicative factor of a certain object In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that Matrices are described by the field of matrix theory. Matrix theory is a branch of Mathematics which focuses on the study of matrices. Matrices can be added, multiplied, and decomposed in various ways, which also makes them a key concept in the field of linear algebra. Linear algebra is the branch of Mathematics concerned with

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 in several different ways Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted

Organization of a 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 Linear algebra, a column vector or column matrix is an m × 1 matrix, i In Linear algebra, a row vector or row matrix is a 1 × n matrix, that is a matrix consisting of a single row \mathbf

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. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Mathematics, and more specifically Set theory, the empty set is the unique set having no ( Zero) members

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

We say that matrix $\mathbf{A}$ is a matrix over the set $\mathbf{S}$ . Cartesian square redirects here For Cartesian squares in Category theory, see Cartesian square (category theory. 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). In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally Since the set of all square matrices over a ring is also a ring, matrix algebra is usually called matrix ring. In Abstract algebra the matrix ring M( n, R) is the set of all n × n matrices over an arbitrary ring

Since this article mainly considers matrices over real numbers, matrices shown here are actually functions $\mathbf{A}\colon \{1, 2, \ldots, m\} \times \{1, 2, \ldots, n\} \to \mathbb{R}.\,\,$

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 $4\times 3$ matrix. In Mathematics, the real numbers may be described informally in several different ways 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 $1\times 9$ matrix, or 9-element row vector.

Basic operations

Sum

Main article: Matrix addition

Two or more matrices of identical dimensions m and n can be added. In Mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together Given m-by-n matrices A and B, their sum A+B is the m-by-n matrix computed by adding corresponding elements:

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

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. In Mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together

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. In Mathematics, scalar multiplication is one of the basic operations defining a Vector space in Linear algebra (or more generally a module in In Mathematics, scalar multiplication is one of the basic operations defining a Vector space in Linear algebra (or more generally a module in In Linear algebra, Real numbers are called Scalars and relate to vectors in a Vector space through the operation of Scalar multiplication e. $(c\mathbf{A})_{i,j} = c \cdot a_{i,j}$ ). For example:

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

Matrix 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 $m\cdot n$ . In Mathematics, the real numbers may be described informally in several different ways 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. In Mathematics, matrix multiplication is the operation of multiplying a matrix with either a scalar or another matrix If A is an m-by-n matrix and B is an n-by-p matrix, then their matrix product AB is the m-by-p matrix given by:

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

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. In Mathematics, commutativity is the ability to change the order of something without changing the end result It may also happen that AB is defined but BA is not defined.

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, matrix multiplication is the operation of multiplying a matrix with either a scalar or another matrix In mathematics the Kronecker product, denoted by \otimes is an operation on two matrices of arbitrary size resulting in a Block matrix.

Linear transformations

Main article: Transformation matrix

Matrices can conveniently represent linear transformations because matrix multiplication neatly corresponds to the composition of maps, as will be described next. In Linear algebra, Linear transformations can be represented by matrices. In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that This same property makes them powerful data structures in high-level programming languages.

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. Basis vector redirects here For basis vector in the context of crystals see Crystal structure.

Ranks

Main article: Rank of a matrix

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. The column rank of a matrix A is the maximal number of Linearly independent columns of A. The column rank of a matrix A is the maximal number of Linearly independent columns of A. In Mathematics, the dimension of a Vector space V is the cardinality (i In Mathematics, the image of a preimage under a given function is the set of all possible function outputs when taking each element of the preimage It can also be defined without reference to linear algebra as follows: the rank of an m-by-n matrix A is the smallest number k such that A can be written as a product BC where B is an m-by-k matrix and C is a k-by-n matrix (although this is not a practical way to compute the rank).

Transpose

Main article: Transpose

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. This article is about the Matrix Transpose operator For other uses see Transposition In Linear algebra, the transpose of a This article is about the Matrix Transpose operator For other uses see Transposition In Linear algebra, the transpose of a 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 Mathematics, any Vector space V has a corresponding dual vector space (or just dual space for short consisting of all Linear functionals

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

Square root

Main article: Square root of a matrix

Given two bounded matrices T and B, B is a square root of T if T = B*B. In Mathematics, the square root of a matrix extends the notion of Square root from numbers to matrices.

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. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real Unless n = 1, this ring is not commutative. In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property

M(n, R), the ring of real square matrices, is a real unitary 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 M(n, C), the ring of complex square matrices, is a complex associative algebra.

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. In Linear algebra, the identity matrix or unit matrix of size n is the n -by- n Square matrix with ones on the Main In Linear algebra, the main diagonal (sometimes leading diagonal or primary diagonal) of a matrix A is the collection of cells A_{ij} For example, if n = 3:

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

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. In Linear algebra, an n -by- n (square matrix A is called invertible or non-singular if there exists an n -by- An n by n matrix A is invertible if and only if there exists a matrix B such that

AB = I_n ( = BA).

In this case, B is the inverse matrix of A, denoted by A⁻¹. In Linear algebra, an n -by- n (square matrix A is called invertible or non-singular if there exists an n -by- 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, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group In Mathematics, the general linear group of degree n is the set of n × n invertible matrices, together with the operation

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. In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes (Eigen means "own" in German and in Dutch. The German language (de ''Deutsch'') is a West Germanic language and one of the world's major languages. Dutch ( is a West Germanic language spoken by around 24 million people 22 million of which are from the Netherlands, Belgium and Suriname ) 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. In Linear algebra, one associates a Polynomial to every Square matrix, its characteristic polynomial. This is a polynomial of degree n and has therefore n complex roots (counting multiple roots according to their multiplicity). In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations In this sense, every square matrix has n complex eigenvalues.

The determinant of a square matrix A is the product of its n eigenvalues, but it can also be defined by the Leibniz formula. In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n In Algebra, the Leibniz formula expresses the Determinant of a square matrix A = (a_{ij}_{ij = 1 \dots n} in terms of permutations of the matrix' Invertible matrices are precisely those matrices with a nonzero determinant.

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 efficient Algorithm for solving systems of linear equations, to find the rank of a matrix In Mathematics, a system of linear equations (or linear system) is a collection of Linear equations involving the same set of Variables For example

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 In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally

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 f(x = \sum_{n=0}^\infty a_n \left( x-c \right^n = a_0 +

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, $a_{i,j}=a_{j,i} \Leftrightarrow \mathbf{A}^\mathrm{T} = \mathbf{A}$ . In Linear algebra, a symmetric matrix is a Square matrix, A, that is equal to its Transpose A = A^{T}
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 Linear algebra, a skew-symmetric (or antisymmetric) matrix is a Square matrix A whose Transpose is also its negative 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. A Hermitian matrix (or self-adjoint matrix) is a Square matrix with complex entries which is equal to its own Conjugate transpose &mdash that 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
Toeplitz matrices have common elements on their diagonals, that is, $\,\! a_{i,j}=a_{i+1,j+1}$ . In the mathematical discipline of Linear algebra, a Toeplitz matrix or diagonal-constant matrix, named after Otto Toeplitz, is a matrix
Stochastic matrices are square matrices whose rows are probability vectors; they are used to define Markov chains. In Mathematics, a stochastic matrix, probability matrix, or transition matrix is used to describe the transitions of a Markov chain. 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 Stochastic process with the Markov property.
A square matrix A is called idempotent if $\mathbf{A}^2=\mathbf{AA}=\mathbf{A}$ . Idempotence ˌaɪdɨmˈpoʊtəns describes the property of operations in Mathematics and Computer science which means that multiple applications of the operation

For a more extensive list see list of matrices. This page lists some important classes of matrices used in Mathematics, Science and Engineering:

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. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real 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 Abstract algebra, one associates to certain objects a ring, the object's 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

Similarly, if the entries are taken from a semiring S, matrix addition and multiplication can still be defined as usual. In Abstract algebra, a semiring is an Algebraic structure similar to a ring, but without the requirement that each element must have an Additive inverse 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 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. In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property In Mathematics, an associative algebra is a Vector space (or more generally a module) which also allows the multiplication of vectors in a distributive 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 Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n In Algebra, the Leibniz formula expresses the Determinant of a square matrix A = (a_{ij}_{ij = 1 \dots n} in terms of permutations of the matrix'

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

Matrices over a polynomial ring are important in the study of control theory. In Mathematics, especially in the field of Abstract algebra, a polynomial ring is a ring formed from the set of Polynomials in one or more variables Control theory is an interdisciplinary branch of Engineering and Mathematics, that deals with the behavior of Dynamical systems The desired output

Matrices without entries

A subtle question that is hardly ever posed is whether there is such a thing as a 3-by-0 matrix. That would be a matrix with 3 rows but without any columns, which seems absurd. However, if one wants to be able to have matrices for all linear maps between finite dimensional vector spaces, one needs such matrices, since there is nothing wrong with linear maps from a 0-dimensional space to a 3-dimensional space (in fact if the spaces are fixed there is one such map, the zero map). So one is led to admit that there is exactly one 3-by-0 matrix (which has 3×0=0 entries; not null entries but none at all). Similarly there are matrices with a positive number of columns but no rows.

Even in absence of entries, one must still keep track of the number of rows and columns, since the product BC where B is the 3-by-0 matrix and C is a 0-by-4 matrix is a perfectly normal 3-by-4 matrix, all of whose 12 entries are 0 (as they are given by an empty sum). In Mathematics, the empty sum, or nullary sum, is the result of adding no numbers in Summation for example Note that this computation of BC justifies the criterion given above for the rank of a matrix in terms of possible expressions as a product: the 3-by-4 matrix with zero entries certainly has rank 0, so it should be the product of a 3-by-0 matrix and a 0-by-4 matrix. ^[1]

To allow and distinguish between matrices without entries, matrices should formally be defined, in a somewhat pedantic computer science style, as quadruples (A, r, c, M), where A is the set in which the entries live, r and c are the (natural) numbers of rows and columns, and M is the rectangular collection of rc elements of A (the matrix in the usual sense).

History

The study of matrices is quite old. A 3-by-3 magic square appears in the Chinese literature dating from as early as 650 BC. In Recreational mathematics, a magic square of order n is an arrangement of n ² numbers usually distinct Integers in a square, such Chinese literature extends back thousands of years from the earliest recorded dynastic court Archives to the mature fictional Novel that arose during the Ming Dynasty ^[2]

Matrices have a long history of application in solving linear equations. A linear equation is an Algebraic equation in which each term is either a Constant or the product of a constant and (the first power of a single Variable 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. Mathematics in China emerged independently by the 11th century BC The Nine Chapters on the Mathematical Art ( is a Chinese Mathematics book composed by several generations of scholars from the 10th&ndash2nd century BC and In Mathematics, a system of linear equations (or linear system) is a collection of Linear equations involving the same set of Variables For example ^[3] In the seventh chapter, "Too much and not enough," the concept of a determinant first appears almost 2000 years before its publication by the Japanese mathematician Seki Kowa in 1683 and the German mathematician Gottfried Leibniz in 1693. In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n In the History of mathematics, Japanese mathematics or wasan (和算 denotes a genuinely distinct kind of mathematics developed in Japan during the or (born 1637/1642? – October 24, 1708) was a Japanese Mathematician who created a new algebraic notation system and laid

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. The araB gene Promoter is a bacterial promoter activated by e L-arabinose binding This article deals with the geophysical region in Asia For geopolitical treatments see South Asia. Indian mathematics &mdashwhich here is the mathematics that emerged in South Asia zero, Negative numbers, Arithmetic, and Algebra. Indian astronomy —the earliest textual mention of which is given in the religious literature of India (2nd millennium BCE—became an established tradition by the 1st millennium BCE Combinatorics is a branch of Pure mathematics concerning the study of discrete (and usually finite) objects 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. Baghdad (بغداد) is the Capital of Iraq and of Baghdad Governorate, with which it is also coterminous Events By Place Asia Wood carvers commissioned by China's Song Dynasty complete a carving of the entire Buddhist canon 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 ^[2]

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. Gabriel Cramer ( July 31, 1704 - January 4, 1752) was a Swiss Mathematician, born in Geneva. Cramer's rule is a Theorem in Linear algebra, which gives the solution of a System of linear equations in terms of Determinants It is named after Year 1750 ( MDCCL) was a Common year starting on Thursday (link will display the full calendar of the Gregorian calendar (or a Carl Friedrich Gauss and Wilhelm Jordan developed Gauss-Jordan elimination in the 1800s. Johann Carl Friedrich Gauss (ˈɡaʊs, Gauß Carolus Fridericus Gauss ( 30 April 1777 – 23 February 1855) was a German Wilhelm Jordan ( March 1 1842 - April 17 1899) was a German Geodesist who did surveys in Germany and Africa and In Linear algebra, Gauss–Jordan elimination is a version of Gaussian elimination that puts zeros both above and below each Pivot element as it goes from

The term "matrix" was coined in 1848 by J. J. Sylvester. Year 1848 ( MDCCCXLVIII) was a Leap year starting on Saturday (link will display the full calendar of the Gregorian Calendar (or a Leap James Joseph Sylvester ( September 3, 1814 London – March 15, 1897 Oxford) was an English Mathematician Cayley, Hamilton, Grassmann, Frobenius and von Neumann are among the famous mathematicians who have worked on matrix theory. Arthur Cayley ( August 16 1821 - January 26 1895) was a British Mathematician. Sir William Rowan Hamilton (4 August 1805 &ndash 2 September 1865 was an Irish Mathematician, Physicist, and Astronomer who Hermann Günther Grassmann ( April 15, 1809, Stettin ( Szczecin) &ndash September 26, 1877, Stettin) was a Ferdinand Georg Frobenius ( October 26, 1849 – August 3, 1917) was a German Mathematician, best-known for his contributions

Olga Taussky-Todd (1906-1995) made important contributions to matrix theory, using it 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 Aeroelasticity is the science which studies the interaction among inertial, elastic, and aerodynamic forces World War II, or the Second World War, (often abbreviated WWII) was a global military conflict which involved a majority of the world's nations, including She has been called "a torchbearer" for matrix theory. ^[4]

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 United States 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. 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 one of the five NCTM -standards-based High school Mathematics Curriculum development projects funded by the They often require the use of graphing calculators such as the TI-83 which can perform complex operations such as matrix inversion very quickly. The TI-83 series of Graphing calculators is 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 cartridges to add BASIC commands for matrices. The HP 9800 was a family of what were initially called programmable Calculators and later Desktop computers made by Hewlett-Packard which replaced their first Some computer languages such as APL, were designed to manipulate matrices, and mathematical programs such as Mathematica, and others are used to aid computing with matrices. Mathematica is a computer program used widely in scientific engineering and mathematical fields Listed here are a number of computer programs used for performing numerical calculations acslX is a software application for modeling and evaluating the performance

Applications

Encryption

See also: Matrix encryption

Matrices can be used to encrypt numerical data. In Classical cryptography, the Hill cipher is a polygraphic substitution cipher based on Linear algebra. 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.

Computer graphics

See also: Transformation matrix

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

References

^ de Boor, Carl (1990), “An empty exercise”, ACM SIGNUM Newsletter 25 (2): 3–7, doi:10.1145/122272.122273, <http://ftp.cs.wisc.edu/Approx/empty.pdf> . This page lists some important classes of matrices used in Mathematics, Science and Engineering: In the mathematical discipline of Linear algebra, a matrix decomposition is a Factorization of a matrix into some Canonical form. A logical matrix or Boolean matrix is a matrix (a rectangular table with entries from the Boolean domain B = {0 1} In Mathematics, matrix calculus is a specialized notation for doing Multivariable calculus, especially over spaces of matrices, where it defines the Carl R de Boor (born 1937 is a German-American mathematician and professor emeritus at the University of Wisconsin-Madison. A digital object identifier ( DOI) is a permanent identifier given to an Electronic document.
Nett, C. N. & Haddad, W. M. (1993), “A system-theoretic appropriate realization of the empty matrix concept”, IEEE Transactions on Automatic Control 38 (5): 771–775, ISSN 0018-9286, DOI 10. An International Standard Serial Number ( ISSN) is a unique eight-digit number used to identify a print or electronic Periodical publication. 1109/9. 277245 .
^ ^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.
^ Ivars Peterson. Ivars Peterson is an award-winning Mathematics writerHe is currently Director of Publications for Journals and Communications at the Mathematical Association Matrices, Circles, and Eigenthings.

External links

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
- Matri-tri-ca - Matrix Calculator
- 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

Imperial College London (officially The Imperial College of Science Technology and Medicine as given in its Royal Charter It is one of only three universities to have reached The University of Minnesota Twin Cities ( U of M or The U) is the oldest and largest part of the University of Minnesota system.

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