In mathematics, and in particular linear algebra, a pseudoinverse of a matrix is a generalization of the inverse matrix. The most widely known type of matrix pseudoinverse is the Moore–Penrose pseudoinverse, which was independently described by E. H. Moore in 1920, Arne Bjerhammar in 1951 and Roger Penrose in 1955. Earlier, 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 pseudoinverse. 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 unique 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 following discussion, the following conventions are adopted.
* will denote one of the fields of real or complex numbers, denoted , respectively. The vector space of matrices over is denoted by .
*For , and denote the transpose and Hermitian transpose (also called conjugate transpose) respectively. If , then .
*For , then denotes the range (image) of (the space spanned by the column vectors of ) and denotes the kernel (null space) of .
*Finally, for any positive integer , denotes the identity matrix.
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