Editing Moore–Penrose inverse (section)

{{Short description|Most widely known generalized inverse of a matrix}}
In [[mathematics]], and in particular [[linear algebra]], the '''Moore–Penrose inverse''' {{tmath| A^+ }} of a [[matrix (mathematics)|matrix]] {{tmath| A }}, often called the '''pseudoinverse''', is the most widely known generalization of the [[inverse matrix]].<ref>{{multiref | {{harvnb|Ben-Israel|Greville|2003|p=7}} | {{harvnb| Campbell|Meyer|1991|p=10}} | {{harvnb|Nakamura|1991|p=42}} | {{harvnb|Rao|Mitra|1971|p=50–51}} }}</ref> It was independently described by [[E. H. Moore]] in 1920,<ref name="Moore1920">{{cite journal | last=Moore | first=E. H. | author-link=E. H. Moore | title=On the reciprocal of the general algebraic matrix | journal=[[Bulletin of the American Mathematical Society]] | volume=26 |issue=9| pages=394–95 | year=1920 | url =http://projecteuclid.org/euclid.bams/1183425340 | doi = 10.1090/S0002-9904-1920-03322-7 | doi-access=free }}</ref> [[Arne Bjerhammar]] in 1951,<ref name="Bjerhammar1951">{{cite journal | last=Bjerhammar| first=Arne| author-link=Arne Bjerhammar | title=Application of calculus of matrices to method of least squares; with special references to geodetic calculations| journal=Trans. Roy. Inst. Tech. Stockholm | year=1951 | volume = 49}}</ref> and [[Roger Penrose]] in 1955.<ref name="Penrose1955">{{cite journal | last=Penrose | first=Roger | author-link=Roger Penrose | title=A generalized inverse for matrices | journal=[[Proceedings of the Cambridge Philosophical Society]] | volume=51 | issue=3 | pages=406–13 | year=1955 | doi=10.1017/S0305004100030401| bibcode=1955PCPS...51..406P | doi-access=free }}</ref> Earlier, [[Erik Ivar Fredholm]] had introduced the concept of a pseudoinverse of [[integral operator]]s 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" ([[Ordinary least squares|least squares]]) approximate solution to a [[system of linear equations]] that lacks an exact solution (see below under [[#Applications|§ Applications]]).
Another use is to find the minimum ([[Euclidean norm|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 for all rectangular matrices whose entries are [[Real number|real]] or [[Complex number|complex]] numbers. Given a rectangular matrix with real or complex entries, its pseudoinverse is unique. 
It can be computed using the [[singular value decomposition]]. In the special case where {{tmath| A}} is a [[normal matrix]] (for example, a Hermitian matrix), the pseudoinverse {{tmath| A^+ }} [[Dual_space#Quotient spaces and annihilators|annihilates]] the [[Kernel (linear algebra)|kernel]] of {{tmath| A}} and acts as a traditional inverse of {{tmath| A}} on the subspace [[Orthogonal complement|orthogonal]] to the kernel.