Matrix (mathematics)From Wikipedia, the free encyclopediaJump to navigationJump to searchFor other uses, seeMatrix."Matrix theory" redirects here. For the physics topic, seeMatrix string theory.Anm×nmatrix: themrows are horizontal and thencolumns are vertical. Each element of a matrix is oftendenoted by a variable with twosubscripts. For example,a2,1represents the element at the second row and firstcolumn of the matrix.Inmathematics, amatrix(pluralmatrices) is arectangulararray[1](seeirregular matrix)ofnumbers,symbols, orexpressions, arranged inrowsandcolumns.[2][3]For example, the dimensionof the matrix below is 2 × 3 (read "two by three"), because there are two rows and three columns:Provided that they have the same size (each matrix has the same number of rows and the samenumber of columns as the other), two matrices can beaddedor subtracted element by element(seeconformable matrix). The rule formatrix multiplication, however, is thattwo matrices can bemultiplied only when the number of columns in the first equals the number of rows in thesecond(i.e., the inner dimensions are the same,nfor an (m×n)-matrix times an (n×p)-matrix,resulting in an (m×p)-matrix). There is no product the other way round, a first hint that matrixmultiplication is notcommutative. Any matrix can bemultipliedelement-wise by ascalarfrom itsassociatedfield.The individual items in anm×nmatrixA, often denoted byai,j, whereiandjusually vary from 1tomandn, respectively, are called itselementsorentries.[4]For conveniently expressing anelement of the results of matrix operations the indices of the element are often attached to theparenthesized or bracketed matrix expression; e.g., (AB)i,jrefers to an element of a matrixproduct. In the context ofabstract index notationthis ambiguously refers also to the whole matrixproduct.A major application of matrices is to representlinear transformations, that is, generalizationsoflinear functionssuch asf(x) = 4x. For example, therotationofvectorsin three-dimensionalspace is a linear transformation, which can be represented by arotation matrixR:ifvis acolumn vector(a matrix with only one column) describing thepositionof a point in space,the productRvis a column vector describing the position of that point after a rotation. Theproduct of twotransformation matricesis a matrix that represents thecompositionoftwotransformations. Another application of matrices is in the solution ofsystems of linearequations. If the matrix issquare, it is possible to deduce some of its properties by computingitsdeterminant. For example, a square matrixhas an inverseif and only ifits determinant isnotzero. Insight into thegeometryof a linear transformation is obtainable (along with otherinformation) from the matrix'seigenvalues and eigenvectors.

Applications of matrices are found in most scientific fields. In every branch ofphysics,includingclassical mechanics,optics,electromagnetism,quantum mechanics, andquantumelectrodynamics

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