In mathematics, and in particular linear algebra, the Moore–Penrose inverse of a matrix , often called the pseudoinverse, 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. 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" (least squares) approximate solution to a system of linear equations that lacks an exact 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 for all rectangular matrices whose entries are real or 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 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:
Note that and are idempotent operators, as follows from and . More specifically, projects onto the image of (equivalently, the span of the rows of ), and projects onto the image of (equivalently, the span of the columns of ). In fact, the above four conditions are fully equivalent to and being such orthogonal projections: projecting onto the image of implies , and projecting onto the image of implies .
The pseudoinverse exists for any matrix . If furthermore is full rank, that is, its rank is , then can be given 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 using the singular value decomposition. Any matrix can be decomposed as for some isometries and diagonal nonnegative real matrix . The pseudoinverse can then be written as , where is the pseudoinverse of and can be obtained by transposing the matrix and replacing the nonzero values with their multiplicative inverses. 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 , 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 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 ; otherwise, , or . The kernel and image of the pseudoinverse coincide with those of the conjugate transpose: and .
Identities The following identity formula can be used to cancel or expand certain subexpressions involving pseudoinverses:
Equivalently, substituting for gives
while substituting for gives
Reduction to Hermitian case The computation of the pseudoinverse is reducible to its construction in the Hermitian case. This is possible through the equivalences:
as and are Hermitian.
Pseudoinverse of products
The equality does not hold in general. Rather, suppose . Then the following are equivalent:
and and
and .
Any of the following is a sufficient condition for : has orthonormal columns (then ), or has orthonormal rows (then ), or has linearly independent columns (then ) and has linearly independent rows (then ), or , or .
The following is a necessary condition for :
The fourth sufficient condition yields the equalities
Here is a counterexample where :
Projectors and are orthogonal projection operators, that is, they are Hermitian (, ) and idempotent ( and ). The following hold: and is the orthogonal projector onto the range of (which equals the orthogonal complement of the kernel of ). is the orthogonal projector onto the range of (which equals the orthogonal complement of the kernel of ). is the orthogonal projector onto the kernel of . is the orthogonal projector onto the kernel of .
The last two properties imply the following identities:
Another property is the following: if is Hermitian and idempotent (true if and only if it represents an orthogonal projection), then, for any matrix the following equation holds:
This can be proven by defining matrices , , and checking that is indeed a pseudoinverse for by verifying that the defining properties of the pseudoinverse hold, when is Hermitian and idempotent.
From the last property it follows that, if is Hermitian and idempotent, for any matrix
Finally, if is an orthogonal projection matrix, then its pseudoinverse trivially coincides with the matrix itself, that is, .
Geometric construction If we view the matrix as a linear map over the field then can be decomposed as follows. We write for the direct sum, for the orthogonal complement, for the kernel of a map, and for the image of a map. Notice that and . The restriction is then an isomorphism. This implies that on is the inverse of this isomorphism, and is zero on
In other words: To find for given in , first project orthogonally onto the range of , finding a point in the range. Then form , that is, find those vectors in that sends to . This will be an affine subspace of parallel to the kernel of . The element of this subspace that has the smallest length (that is, is closest to the origin) is the answer we are looking for. It can be found by taking an arbitrary member of and projecting it orthogonally onto the orthogonal complement of the kernel of .
This description is closely related to the minimum-norm solution to a linear system.
Limit relations The pseudoinverse are limits:
(see Tikhonov regularization). These limits exist even if or do not exist.
Continuity In contrast to ordinary matrix inversion, the process of taking pseudoinverses is not continuous: if the sequence converges to the matrix (in the maximum norm or Frobenius norm, say), then need not converge to . However, if all the matrices have the same rank as , will converge to .
Derivative Let be a real-valued differentiable matrix function with constant rank in a neighborhood of a point . The derivative of at may be calculated in terms of the derivative of at :
where the functions , and derivatives on the right side are evaluated at (that is, , , etc.). For a complex matrix, the transpose is replaced with the conjugate transpose. For a real-valued symmetric matrix, the Magnus-Neudecker derivative is established.
Examples Since for invertible matrices the pseudoinverse equals the usual inverse, only examples of non-invertible matrices are considered below.
For the pseudoinverse is The uniqueness of this pseudoinverse can be seen from the requirement , since multiplication by a zero matrix would always produce a zero matrix. For the pseudoinverse is . Indeed, and thus Similarly, and thus Note that is neither injective nor surjective, and thus the pseudoinverse cannot be computed via nor as and are both singular, and furthermore is neither a left nor a right inverse. Nonetheless, the pseudoinverse can be computed via SVD observing that and thus For For The denominators are here For For the pseudoinverse is For this matrix, the left inverse exists and thus equals indeed,
Special cases Scalars It is also possible to define a pseudoinverse for scalars and vectors. This amounts to treating these as matrices. The pseudoinverse of a scalar is zero if is zero and the reciprocal of otherwise:
Vectors The pseudoinverse of the null (all zero) vector is the transposed null vector. The pseudoinverse of a non-null vector is the conjugate transposed vector divided by its squared magnitude:
Diagonal matrices 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, .
Linearly independent columns If the rank of is identical to the number of columns, , (for ,) there are linearly independent columns, and is invertible. In this case, an explicit formula is:
It follows that is then a left inverse of : .
Linearly independent rows If the rank of is identical to the number of rows, , (for ,) there are linearly independent rows, and is invertible. In this case, an explicit formula is:
It follows that is a right inverse of : .
Orthonormal columns or rows This is a special case of either full column rank or full row rank (treated above). If has orthonormal columns () or orthonormal rows (), then:
Normal matrices If is normal, that is, it commutes with its conjugate transpose, then its pseudoinverse can be computed by diagonalizing it, mapping all nonzero eigenvalues to their inverses, and mapping zero eigenvalues to zero. A corollary is that commuting with its transpose implies that it commutes with its pseudoinverse.
EP matrices A (square) matrix is said to be an EP matrix if it commutes with its pseudoinverse. In such cases (and only in such cases), it is possible to obtain the pseudoinverse as a polynomial in . A polynomial such that can be easily obtained from the characteristic polynomial of or, more generally, from any annihilating polynomial of .
Orthogonal projection matrices This is a special case of a normal matrix with eigenvalues 0 and 1. If is an orthogonal projection matrix, that is, and , then the pseudoinverse trivially coincides with the matrix itself:
Circulant matrices For a circulant matrix , the singular value decomposition is given by the Fourier transform, that is, the singular values are the Fourier coefficients. Let be the Discrete Fourier Transform (DFT) matrix; then