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 , the transpose is denoted and the Hermitian transpose (also called conjugate transpose) is denoted . 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 . For any positive integer , the identity matrix is denoted .
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:
These conditions are equivalent to being the projection onto the support of , and being the projection onto the image of . The pseudoinverse exists for any matrix . If furthermore is full rank, that is, its rank is , then has a particularly simple algebraic expression.
In particular, when has linearly independent columns (equivalently, is injective, and thus is invertible), can be computed asThis particular pseudoinverse is a left inverse, that is, . If on the other hand has linearly independent rows (equivalently, is surjective, and thus is invertible), can be computed asThis is a right inverse, as .
In the more general case, the pseudoinverse can be expressed leveraging the singular value decomposition. Any matrix can be decomposed as for some isometries and diagonal positive real matrix . The pseudoinverse can then be written as . That this matrix satisfies the above requirement is directly verified observing that and , which are the projections onto image and support of , respectively.
Properties
Existence and uniqueness
As discussed above, for any matrix there is one and only one pseudoinverse .
A matrix satisfying only the first of the conditions given above, namely , it is known as a generalized inverse. If the matrix also satisfies the second condition, namely , it is called a generalized reflexive inverse. Generalized inverses always exist but are not in general unique. Uniqueness is a consequence of the last two conditions.
Basic properties
Proofs for the properties below can be found at b:Topics in Abstract Algebra/Linear algebra. If has real entries, then so does . If is invertible, its pseudoinverse is its inverse. That is, . The pseudoinverse of a squared diagonal matrix is obtained by taking the reciprocal of the nonzero diagonal elements. Formally, if is a squared diagonal matrix with and , then . More generally, if is any rectangular matrix whose only nonzero elements are on the diagonal, meaning , , then is a rectangular matrix whose diagonal elements are the reciprocal of the original ones, that is, . The pseudoinverse of the pseudoinverse is the original matrix: . Pseudoinversion commutes with transposition, complex conjugation, and taking the conjugate transpose: The pseudoinverse of a scalar multiple of is the reciprocal multiple of : for .
Identities
The following identity formula can be used to cancel or expand certain subexpressions involving pseudoinverses: