In mathematics, and in particular linear algebra, the Moore–Penrose inverse of a matrix 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 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. Notation 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 , (standing for "range") denotes the column space (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. Definition For , a pseudoinverse of is defined as a matrix satisfying all of the following four criteria, known as the Moore–Penrose conditions: need not be the general identity matrix, but it maps all column vectors of to themselves: acts like a weak inverse: is Hermitian: is also Hermitian: exists for any matrix , but, when the latter has full rank (that is, the rank of is ), then can be expressed as a simple algebraic formula. In particular, when has linearly independent columns (and thus matrix is invertible), can be computed as This particular pseudoinverse constitutes a left inverse,