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.[1] It was independently described by E. H. Moore in 1920,[2] Arne Bjerhammar in 1951,[3] and Roger Penrose in 1955.[4] 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 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
editIn 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
editFor , a pseudoinverse of A is defined as a matrix
satisfying all of the following four criteria, known as the Moore–Penrose conditions:[4][5]
need not be the general identity matrix, but it maps all column vectors of A 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 as
This 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 as
This 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 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.[6] That this matrix satisfies the above requirement is directly verified observing that
and
, which are the projections onto image and support of
, respectively.
Properties
editExistence and uniqueness
editAs discussed above, for any matrix there is one and only one pseudoinverse
.[5]
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
editProofs 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,
.[7]: 243
- The pseudoinverse of the pseudoinverse is the original matrix:
.[7]: 245
- Pseudoinversion commutes with transposition, complex conjugation, and taking the conjugate transpose:[7]: 245
- The pseudoinverse of a scalar multiple of
is the reciprocal multiple of
:
for
.
- The kernel and image of the pseudoinverse coincide with those of the conjugate transpose:
and
.
Identities
editThe 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
editThe 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
editThe equality does not hold in general. Rather, suppose
. Then the following are equivalent:[8]
The following are sufficient conditions 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
edit 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
.[5]
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:[9]
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
editIf 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
editThe pseudoinverse are limits: (see Tikhonov regularization). These limits exist even if
or
do not exist.[5]: 263
Continuity
editIn 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
.[10]
Derivative
editLet be a real-valued differentiable matrix function with constant rank at a point
.The derivative of
at
may be calculated in terms of the derivative of
at
:[11]
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.[12] For a real-valued symmetric matrix, the Magnus-Neudecker derivative is established.[13]
Examples
editSince 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
editScalars
editIt 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
editThe 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
editThe 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
editIf the rank of is identical to its column rank,
, (for
,) there are
linearly independent columns, and
is invertible. In this case, an explicit formula is:[14]
It follows that is then a left inverse of
:
.
Linearly independent rows
editIf the rank of is identical to its row rank,
, (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
editThis 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
editIf 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
editA (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
.[15]
Orthogonal projection matrices
editThis 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
editFor 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[16]
Construction
editRank decomposition
editLet denote the rank of
. Then
can be (rank) decomposed as
where
and
are of rank
. Then
.
The QR method
editFor computing the product
or
and their inverses explicitly is often a source of numerical rounding errors and computational cost in practice. An alternative approach using the QR decomposition of
may be used instead.
Consider the case when is of full column rank, so that
. Then the Cholesky decomposition
, where
is an upper triangular matrix, may be used. Multiplication by the inverse is then done easily by solving a system with multiple right-hand sides,
which may be solved by forward substitution followed by back substitution.
The Cholesky decomposition may be computed without forming explicitly, by alternatively using the QR decomposition of
, where
has orthonormal columns,
, and
is upper triangular. Then
so is the Cholesky factor of
.
The case of full row rank is treated similarly by using the formula and using a similar argument, swapping the roles of
and
.
Using polynomials in matrices
editFor an arbitrary , one has that
is normal and, as a consequence, an EP matrix. One can then find a polynomial
such that
. In this case one has that the pseudoinverse of
is given by[15]
Singular value decomposition (SVD)
editA computationally simple and accurate way to compute the pseudoinverse is by using the singular value decomposition.[14][5][17] If is the singular value decomposition of
, then
. For a rectangular diagonal matrix such as
, we get the pseudoinverse by taking the reciprocal of each non-zero element on the diagonal, leaving the zeros in place, and then transposing the matrix. In numerical computation, only elements larger than some small tolerance are taken to be nonzero, and the others are replaced by zeros. For example, in the MATLAB or GNU Octave function pinv, the tolerance is taken to be t = ε⋅max(m, n)⋅max(Σ), where ε is the machine epsilon.
The computational cost of this method is dominated by the cost of computing the SVD, which is several times higher than matrix–matrix multiplication, even if a state-of-the art implementation (such as that of LAPACK) is used.
The above procedure shows why taking the pseudoinverse is not a continuous operation: if the original matrix has a singular value 0 (a diagonal entry of the matrix
above), then modifying
slightly may turn this zero into a tiny positive number, thereby affecting the pseudoinverse dramatically as we now have to take the reciprocal of a tiny number.
Block matrices
editOptimized approaches exist for calculating the pseudoinverse of block-structured matrices.
The iterative method of Ben-Israel and Cohen
editAnother method for computing the pseudoinverse (cf. Drazin inverse) uses the recursion
which is sometimes referred to as hyper-power sequence. This recursion produces a sequence converging quadratically to the pseudoinverse of if it is started with an appropriate
satisfying
. The choice
(where
, with
denoting the largest singular value of
)[18] has been argued not to be competitive to the method using the SVD mentioned above, because even for moderately ill-conditioned matrices it takes a long time before
enters the region of quadratic convergence.[19] However, if started with
already close to the Moore–Penrose inverse and
, for example
, convergence is fast (quadratic).
Updating the pseudoinverse
editFor the cases where has full row or column rank, and the inverse of the correlation matrix (
for
with full row rank or
for full column rank) is already known, the pseudoinverse for matrices related to
can be computed by applying the Sherman–Morrison–Woodbury formula to update the inverse of the correlation matrix, which may need less work. In particular, if the related matrix differs from the original one by only a changed, added or deleted row or column, incremental algorithms exist that exploit the relationship.[20][21]
Similarly, it is possible to update the Cholesky factor when a row or column is added, without creating the inverse of the correlation matrix explicitly. However, updating the pseudoinverse in the general rank-deficient case is much more complicated.[22][23]
Software libraries
editHigh-quality implementations of SVD, QR, and back substitution are available in standard libraries, such as LAPACK. Writing one's own implementation of SVD is a major programming project that requires a significant numerical expertise. In special circumstances, such as parallel computing or embedded computing, however, alternative implementations by QR or even the use of an explicit inverse might be preferable, and custom implementations may be unavoidable.
The Python package NumPy provides a pseudoinverse calculation through its functions matrix.I
and linalg.pinv
; its pinv
uses the SVD-based algorithm. SciPy adds a function scipy.linalg.pinv
that uses a least-squares solver.
The MASS package for R provides a calculation of the Moore–Penrose inverse through the ginv
function.[24] The ginv
function calculates a pseudoinverse using the singular value decomposition provided by the svd
function in the base R package. An alternative is to employ the pinv
function available in the pracma package.
The Octave programming language provides a pseudoinverse through the standard package function pinv
and the pseudo_inverse()
method.
In Julia (programming language), the LinearAlgebra package of the standard library provides an implementation of the Moore–Penrose inverse pinv()
implemented via singular-value decomposition.[25]
Applications
editLinear least-squares
editThe pseudoinverse provides a least squares solution to a system of linear equations.[26]For , given a system of linear equations
in general, a vector that solves the system may not exist, or if one does exist, it may not be unique. More specifically, a solution exists if and only if
is in the image of
, and is unique if and only if
is injective. The pseudoinverse solves the "least-squares" problem as follows:
, we have
where
and
denotes the Euclidean norm. This weak inequality holds with equality if and only if
for any vector
; this provides an infinitude of minimizing solutions unless
has full column rank, in which case
is a zero matrix.[27] The solution with minimum Euclidean norm is
[27]
This result is easily extended to systems with multiple right-hand sides, when the Euclidean norm is replaced by the Frobenius norm. Let .
, we have
where
and
denotes the Frobenius norm.
Obtaining all solutions of a linear system
editIf the linear system
has any solutions, they are all given by[28]
for arbitrary vector . Solution(s) exist if and only if
.[28] If the latter holds, then the solution is unique if and only if
has full column rank, in which case
is a zero matrix. If solutions exist but
does not have full column rank, then we have an indeterminate system, all of whose infinitude of solutions are given by this last equation.
Minimum norm solution to a linear system
editFor linear systems with non-unique solutions (such as under-determined systems), the pseudoinverse may be used to construct the solution of minimum Euclidean norm
among all solutions.
- If
is satisfiable, the vector
is a solution, and satisfies
for all solutions.
This result is easily extended to systems with multiple right-hand sides, when the Euclidean norm is replaced by the Frobenius norm. Let .
- If
is satisfiable, the matrix
is a solution, and satisfies
for all solutions.
Condition number
editUsing the pseudoinverse and a matrix norm, one can define a condition number for any matrix:
A large condition number implies that the problem of finding least-squares solutions to the corresponding system of linear equations is ill-conditioned in the sense that small errors in the entries of can lead to huge errors in the entries of the solution.[29]
Generalizations
editIn order to solve more general least-squares problems, one can define Moore–Penrose inverses for all continuous linear operators between two Hilbert spaces
and
, using the same four conditions as in our definition above. It turns out that not every continuous linear operator has a continuous linear pseudoinverse in this sense.[29] Those that do are precisely the ones whose range is closed in
.
A notion of pseudoinverse exists for matrices over an arbitrary field equipped with an arbitrary involutive automorphism. In this more general setting, a given matrix doesn't always have a pseudoinverse. The necessary and sufficient condition for a pseudoinverse to exist is that , where
denotes the result of applying the involution operation to the transpose of
. When it does exist, it is unique.[30] Example: Consider the field of complex numbers equipped with the identity involution (as opposed to the involution considered elsewhere in the article); do there exist matrices that fail to have pseudoinverses in this sense? Consider the matrix
. Observe that
while
. So this matrix doesn't have a pseudoinverse in this sense.
In abstract algebra, a Moore–Penrose inverse may be defined on a *-regular semigroup. This abstract definition coincides with the one in linear algebra.
See also
editNotes
editReferences
edit- Ben-Israel, Adi; Greville, Thomas N.E. (2003). Generalized inverses: Theory and applications (2nd ed.). New York, NY: Springer. doi:10.1007/b97366. ISBN 978-0-387-00293-4.
- Campbell, S. L.; Meyer, C. D. Jr. (1991). Generalized Inverses of Linear Transformations. Dover. ISBN 978-0-486-66693-8.
- Nakamura, Yoshihiko (1991). Advanced Robotics: Redundancy and Optimization. Addison-Wesley. ISBN 978-0201151985.
- Rao, C. Radhakrishna; Mitra, Sujit Kumar (1971). Generalized Inverse of Matrices and its Applications. New York: John Wiley & Sons. p. 240. ISBN 978-0-471-70821-6.
External links
edit- Pseudoinverse at PlanetMath.
- Interactive program & tutorial of Moore–Penrose Pseudoinverse
- Moore–Penrose generalized inverse at PlanetMath.
- Weisstein, Eric W. "Pseudoinverse". MathWorld.
- Weisstein, Eric W. "Moore–Penrose Inverse". MathWorld.
- The Moore–Penrose Pseudoinverse. A Tutorial Review of the Theory
- Online Moore–Penrose Inverse calculator