- Matrix theory
**Matrix theory**is a branch ofmathematics which focuses on the study of matrices. Initially a sub-branch oflinear algebra , it has grown to cover subjects related tograph theory ,algebra ,combinatorics , andstatistics as well.**History**The term "matrix" was first coined in 1848 by

J.J. Sylvester as a name of an array of numbers. In 1855,Arthur Cayley introduced matrix as a representation oflinear transformation . This period was considered as the beginning oflinear algebra and matrix theory. The motivation for linear algebra, and the first use of matrices, was the study ofsystems of linear equations . Related concepts such asdeterminant andGaussian elimination , which existed long before the introduction of matrices, are now part of matrix theory.**Applications**The study of

vector space overfinite field , a branch of linear algebra which is useful incoding theory , naturally leads to the study and use of matrices over finite field in coding theory.Modules are generalizations of vector spaces. They are similar to vector spaces, but defined over rings rather than fields. This leads to the study of matrices over rings. Matrix theory in this area is not often considered as a branch of linear algebra. Among the results listed in Useful theorems, the

Cayley-Hamilton Theorem is valid if the underlying ring iscommutative ,Smith normal form is valid if the underlying ring is aprincipal ideal domain , but others are valid for only matrices overcomplex number s orreal number s.Magic square s andLatin square s, two ancient branches of recreational mathematics, are now reformulated using the language of matrices. The link between Latin squares and coding theory demonstrates that this is not merely a coincidence.With the advance of

computer technology, it is now possible to solve systems of large numbers of linear equations in practice, not just in theory.John von Neumann andHerman Goldstine introducedcondition number s in analyzinground-off error s in 1947. Later, different techniques to calculation, multiplication or factorization of matrices were invented, such as theFast Fourier Transform .The

payoff matrix ingame theory , also introduced by John von Neumann, might be the first application of matrices toeconomics .The

simplex algorithm , a technique involving the operations of matrices of very large size, is used to solveoperations research problems, a field strongly related to economics.Flow network problems, part of bothgraph theory andlinear programming , can be solved using the simplex algorithm, although there are other more efficient methods. Matrices appear elsewhere in graph theory as well; for example, theadjacency matrix representation of a directed orundirected graph . Important matrices incombinatorics are permutation matrices, which representpermutation s, andHadamard matrices .Both adjacency matrices of graphs and permutation matrices are examples of nonnegative matrices, which also include stochastic and doubly stochastic matrices. Stochastic matrices are useful in the study of

stochastic process es, inprobability theory and instatistics . The evaluation of an enormous stochastic matrix is the central idea behind thePageRank algorithm used byGoogle . Each doubly stochastic matrix is aconvex combination of permutation matrices.Another important tool in statistics is the

correlation matrix .For optimization problems involving multi-variable real-value functions, Positive-definite matrices occur in the search for

maxima and minima .There are also practical uses for matrices over arbitrary rings (see

Matrix ring ). In particular, matrices overpolynomial ring s are used incontrol theory .On the pure mathematics side, matrix rings can provide a rich field of counterexamples for mathematical conjectures, amongst other uses. The square matrices also plays a special role, because the "n"×"n" matrices for fixed "n" have many closure properties.

**Useful theorems***

Cayley–Hamilton theorem

* Jordan decomposition

*QR decomposition

*Schur triangulation

*Singular value decomposition

*Smith normal form **ee also***

List of matrices . This list is a rich source of information and links to a very wide variety of matrices from mathematics, science and engineering.

*Real matrices (2 x 2) shows that, when non-singular, a 2 x 2 real matrix is proportional to ashear mapping , asqueeze mapping , or arotation .**References**

***Beezer, Rob**, [*http://linear.ups.edu/index.html "A First Course in Linear Algebra"*] , licensed under GFDL.

***Jim Hefferon**: " [*http://joshua.smcvt.edu/linalg.html/ Linear Algebra*] " (Online textbook)**External links*** [

*http://darkwing.uoregon.edu/~vitulli/441.sp04/LinAlgHistory.html A Brief History of Linear Algebra and Matrix Theory*]

*Wikimedia Foundation.
2010.*