In mathematics, a matrix (plural matrices, or less commonly matrixes) is a rectangular array of numbers, such as

An item in a matrix is called an entry or an element. The example has entries 1, 9, 13, 20, 55, and 6. Entries are often denoted by a variable with two subscripts, as shown on the right; thus in the example above, a2,1 = 20. Matrices of the same size can be added and subtracted entrywise and matrices of compatible sizes can be multiplied. These operations have many of the properties of ordinary arithmetic, except that matrix multiplication is not commutative, that is, AB and BA are not equal in general. Matrices consisting of only one column or row define the components of vectors, while higher-dimensional (e.g., three-dimensional) arrays of numbers define the components of a generalization of a vector called a tensor. Matrices with entries in other fields or rings are also studied.
Matrices are a key tool in linear algebra. One use of matrices is to represent linear transformations, which are higher-dimensional analogs of linear functions of the form f(x) = cx, where c is a constant; matrix multiplication corresponds to composition of linear transformations. Matrices can also keep track of the coefficients in a system of linear equations. For a square matrix, the determinant and inverse matrix (when it exists) govern the behavior of solutions to the corresponding system of linear equations, and eigenvalues and eigenvectors provide insight into the geometry of the associated linear transformation.

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Matrices find many applications. Physics makes use of matrices in various domains, for example in geometrical optics and matrix mechanics; the latter led to studying in more detail matrices with an infinite number of rows and columns. Graph theory uses matrices to keep track of distances between pairs of vertices in a graph. Computer graphics uses matrices to project 3-dimensional space onto a 2-dimensional screen. Matrix calculus generalizes classical analytical notions such as derivatives of functions or exponentials to matrices. The latter is a recurring need in solving ordinary differential equations. Serialism and dodecaphonism are musical movements of the 20th century that use a square mathematical matrix to determine the pattern of music intervals.
A major branch of numerical analysis is devoted to the development of efficient algorithms for matrix computations, a subject that is centuries old but still an active area of research. Matrix decomposition methods simplify computations, both theoretically and practically. For sparse matrices, specifically tailored algorithms can provide speedups; such matrices arise in the finite element method, for example.


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