Arthur Cayley introduced matrix multiplication and the inverse matrix in , making possible the general linear group. The mechanism of group representation became available for describing complex and hypercomplex numbers. Crucially, Cayley used a single letter to denote a matrix, thus treating a matrix as an aggregate object. He also realized the connection between matrices and determinants, and wrote "There would be many things to say about this theory of matrices which should, it seems to me, precede the theory of determinants".
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Jump to navigation Jump to search Numerical linear algebra is the study of how matrix operations can be used to create computer algorithms which efficiently and accurately provide approximate answers to mathematical questions. It is a subfield of numerical analysis , and a type of linear algebra.
Because computers use floating-point arithmetic , they cannot exactly represent irrational data, and many algorithms increase that imprecision when implemented by a computer.
Numerical linear algebra uses properties of vectors and matrices to develop computer algorithms that minimize computer error while retaining efficiency and precision. Numerical linear algebra aims to solve problems of continuous mathematics using finite precision computers, so its applications to the natural and social sciences are as vast as the applications of continuous mathematics.
It is often a fundamental part of engineering and computational science problems, such as image and signal processing , telecommunication , computational finance , materials science simulations, structural biology , data mining , bioinformatics , and fluid dynamics.
Matrix methods are particularly used in finite difference methods , finite element methods , and the modeling of differential equations. Noting the broad applications of numerical linear algebra, Lloyd N. Trefethen and David Bau, III argue that it is "as fundamental to the mathematical sciences as calculus and differential equations",  :x even though it is a comparatively small field.
Numerical linear algebra is centrally concerned with developing algorithms that do not introduce errors when applied to real data on a finite precision computer.
Often this is achieved by iterative methods rather than direct methods.
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