Moore–Penrose inverse
Most widely known generalized inverse of a matrix / From Wikipedia, the free encyclopedia
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In mathematics, and in particular linear algebra, the Moore–Penrose inverse of a matrix , often called the pseudoinverse, is the most widely known generalization of the inverse matrix.[1] It was independently described by E. H. Moore in 1920,[2] Arne Bjerhammar in 1951,[3] and Roger Penrose in 1955.[4] Earlier, Erik Ivar Fredholm had introduced the concept of a pseudoinverse of integral operators in 1903. The terms pseudoinverse and generalized inverse are sometimes used as synonyms for the Moore–Penrose inverse of a matrix, but sometimes applied to other elements of algebraic structures which share some but not all properties expected for an inverse element.
A common use of the pseudoinverse is to compute a "best fit" (least squares) approximate solution to a system of linear equations that lacks an exact solution (see below under § Applications). Another use is to find the minimum (Euclidean) norm solution to a system of linear equations with multiple solutions. The pseudoinverse facilitates the statement and proof of results in linear algebra.
The pseudoinverse is defined and unique for all matrices whose entries are real or complex numbers. It can be computed using the singular value decomposition. In the special case where is a normal matrix (for example, a Hermitian matrix), the pseudoinverse annihilates the kernel of and acts as a traditional inverse of on the subspace orthogonal to the kernel.