Moore–Penrose inverse

In mathematics, and in particular linear algebra, the Moore–Penrose inverse ${\displaystyle A^{+}}$ of a matrix ${\displaystyle A}$ is the most widely known generalization of the inverse matrix. It was independently described by E. H. Moore in 1920, Arne Bjerhammar in 1951, and Roger Penrose in 1955. Earlier, Erik Ivar Fredholm had introduced the concept of a pseudoinverse of integral operators in 1903. When referring to a matrix, the term pseudoinverse, without further specification, is often used to indicate the Moore–Penrose inverse. The term generalized inverse is sometimes used as a synonym for pseudoinverse.

A common use of the pseudoinverse is to compute a "best fit" (least squares) solution to a system of linear equations that lacks a 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 ${\displaystyle A}$ is a normal matrix (for example, a Hermitian matrix), the pseudoinverse ${\displaystyle A^{+}}$ annihilates the kernel of ${\displaystyle A}$ and acts as a traditional inverse of ${\displaystyle A}$ on the subspace orthogonal to the kernel.