View source for Moore–Penrose inverse

{{Short description|Most widely known generalized inverse of a matrix}}
In [[Mathematics|mathematics]], and in particular [[Linear algebra|linear algebra]], the '''Moore–Penrose inverse''' {{tmath| A^+ }} of a [[Matrix (mathematics)|matrix]] {{tmath| A }} is the most widely known [[Generalization|generalization]] of the inverse matrix.{{sfn|Ben-Israel|Greville|2003|p=7}}{{sfn|Campbell|Meyer|1991|p=10}}{{sfn|Nakamura|1991|p=42}}{{sfn|Rao|Mitra|1971|p=50–51}} It was independently described by [[Biography:E. H. Moore|E. H. Moore]]<ref name="Moore1920">{{cite journal | last=Moore | first=E. H. |  title=On the reciprocal of the general algebraic matrix | journal=[[Organization:Bulletin of the American Mathematical Society|Bulletin of the American Mathematical Society]] | volume=26 |issue=9| pages=394–95 | year=1920 | url =http://projecteuclid.org/euclid.bams/1183425340 | doi = 10.1090/S0002-9904-1920-03322-7 | doi-access=free }}</ref> in 1920, Arne Bjerhammar<ref name="Bjerhammar1951">{{cite journal | last=Bjerhammar| first=Arne| author-link=Arne Bjerhammar | title=Application of calculus of matrices to method of least squares; with special references to geodetic calculations| journal=Trans. Roy. Inst. Tech. Stockholm | year=1951 | volume = 49}}</ref> in 1951, and [[Biography:Roger Penrose|Roger Penrose]]<ref name="Penrose1955">{{cite journal | last=Penrose | first=Roger | author-link=Roger Penrose | title=A generalized inverse for matrices | journal=Proceedings of the Cambridge Philosophical Society | volume=51 | issue=3 | pages=406–13 | year=1955 | doi=10.1017/S0305004100030401| bibcode=1955PCPS...51..406P | doi-access=free }}</ref> in 1955. Earlier, [[Biography:Erik Ivar Fredholm|Erik Ivar Fredholm]] had introduced the concept of a pseudoinverse of [[Integral operator|integral operator]]s 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|generalized inverse]] is sometimes used as a synonym for pseudoinverse.

A common use of the pseudoinverse is to compute a "best fit" ([[Ordinary least squares|least squares]]) solution to a [[System of linear equations|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 number|real]] or [[Complex number|complex]] numbers. It can be computed using the [[Singular value decomposition|singular value decomposition]]. In the special case where {{tmath| A}} is a [[Normal matrix|normal matrix]] (for example, a Hermitian matrix), the pseudoinverse {{tmath| A^+ }} annihilates the [[Kernel (linear algebra)|kernel]] of {{tmath| A}} and acts as a traditional inverse of {{tmath| A}} on the subspace orthogonal to the kernel.

==Notation==
In the following discussion, the following conventions are adopted.

* {{tmath| \mathbb{K} }} will denote one of the [[Field (mathematics)|fields]] of real or complex numbers, denoted {{tmath| \mathbb{R} }}, {{tmath| \mathbb{C} }}, respectively. The vector space of {{tmath| m \times n }} matrices over {{tmath| \mathbb{K} }} is denoted by {{tmath| \mathbb{K}^{m \times n} }}.
* For {{tmath| A \in \mathbb{K}^{m\times n} }}, the transpose is denoted {{tmath| A^\operatorname{T} }} and the Hermitian transpose (also called [[Conjugate transpose|conjugate transpose]]) is denoted {{tmath| A^* }}. If <math>\mathbb{K} = \mathbb{R}</math>, then <math>A^* = A^\operatorname{T}</math>.
* For {{tmath| A \in \mathbb{K}^{m\times n} }}, {{tmath| \operatorname{ran}(A) }} (standing for "[[Range of a function|range]]") denotes the [[Column space|column space]] ([[Image (mathematics)|image]]) of {{tmath| A }} (the space spanned by the column vectors of {{tmath| A }}) and {{tmath| \ker(A) }} denotes the [[Kernel (linear algebra)|kernel]] (null space) of {{tmath| A }}.
* For any positive integer {{tmath| n }}, the {{tmath| n \times n }} [[Identity matrix|identity matrix]] is denoted {{tmath| I_n \in \mathbb{K}^{n\times n} }}.

==Definition==
For <math>A \in \mathbb{K}^{m\times n}</math>, a pseudoinverse of {{mvar| A}} is defined as a matrix {{tmath| A^+ \in \mathbb{K}^{n\times m} }} satisfying all of the following four criteria, known as the Moore–Penrose conditions:<ref name="Penrose1955"/><ref name="GvL1996">{{cite book | last=Golub | first=Gene H. | author2=Charles F. Van Loan | title=Matrix computations | url=https://archive.org/details/matrixcomputatio00golu_910 | url-access=limited | edition=3rd | publisher=Johns Hopkins | location=Baltimore | year=1996 | isbn=978-0-8018-5414-9 | pages = [https://archive.org/details/matrixcomputatio00golu_910/page/n283 257]–258| }}</ref>
# {{tmath| A A^+ }} need not be the general identity matrix, but it maps all column vectors of {{mvar| A }} to themselves: <math display="block">A A^+ A = \; A.</math>
# {{tmath| A^+ }} acts like a [[Weak inverse|weak inverse]]: <math display="block">A^+ A A^+ = \; A^+.</math>
# {{tmath| A A^+ }} is [[Hermitian matrix|Hermitian]]: <math display="block">\left(A A^+\right)^* = \; A A^+.</math>
# {{tmath| A^+A }} is also Hermitian: <math display="block">\left(A^+ A\right)^* = \; A^+ A.</math>

Note that <math>A^+A</math> and <math>AA^+</math> are orthogonal projection operators, as follows from <math>(AA^+)^2=A A^+</math> and <math>(A^+ A)^2=A^+ A</math>. More specifically, <math>A^+A</math> projects onto the image of <math>A^T</math> (equivalently, the span of the rows of <math>A</math>), and <math>AA^+</math> projects onto the image of <math>A</math> (equivalently, the span of the columns of <math>A</math>). In fact, the above four conditions are fully equivalent to <math>A^+A</math> and <math>AA^+</math> being such orthogonal projections: <math>AA^+</math> projecting onto the image of <math>A</math> implies <math>(A A^+)A=A</math>, and <math>A^+A</math> projecting onto the image of <math>A^T</math> implies <math>(A^+A)A^+=A^+</math>.

The pseudoinverse <math>A^+</math> exists for any matrix <math>A \in \mathbb{K}^{m\times n}</math>. If furthermore <math>A</math> is full [[Rank (linear algebra)|rank]], that is, its rank is {{tmath| \min \{ m,n \} }}, then {{tmath| A^+ }} can be given a particularly simple algebraic expression. In particular:

* When {{tmath| A }} has linearly independent columns (equivalently, {{tmath| A }} is injective, and thus {{tmath| A^* A }} is invertible), {{tmath| A^+ }} can be computed as<math display="block">A^+ = \left(A^* A\right)^{-1} A^*.</math>This particular pseudoinverse is a ''left inverse'', that is, <math>A^+A = I</math>.
* If, on the other hand, <math>A</math> has linearly independent rows (equivalently, <math>A</math> is surjective, and thus {{tmath| A A^* }} is invertible), {{tmath| A^+ }} can be computed as<math display="block">A^+ = A^* \left(A A^*\right)^{-1}.</math>This is a ''right inverse'', as <math>A A^+ = I</math>.

In the more general case, the pseudoinverse can be expressed leveraging the [[Singular value decomposition|singular value decomposition]]. Any matrix can be decomposed as <math> A=UDV^*</math> for some isometries <math>U,V</math> and diagonal positive real matrix <math>D</math>. The pseudoinverse can then be written as <math>A^+=V D^{-1} U^*</math>. That this matrix satisfies the above requirement is directly verified observing that <math>AA^+=UU^*</math> and <math>A^+ A=VV^*</math>, which are the projections onto image and support of <math>A</math>, respectively.

==Properties==

===Existence and uniqueness===
As discussed above, for any matrix {{tmath| A }} there is one and only one pseudoinverse {{tmath| A^+ }}.<ref name="GvL1996"/>

A matrix satisfying only the first of the conditions given above, namely <math display="inline">A A^+ A = A</math>, it is known as a generalized inverse. If the matrix also satisfies the second condition, namely <math display="inline">A^+ A A^+ = A^+</math>, it is called a [[Generalized inverse#Types of generalized inverses|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 {{tmath| A }} has real entries, then so does {{tmath| A^+ }}.
* If {{tmath| A }} is [[Invertible matrix|invertible]], its pseudoinverse is its inverse. That is, <math>A^+ = A^{-1}</math>.<ref name="SB2002">{{Cite book | last1=Stoer | first1=Josef | last2=Bulirsch | first2=Roland | title=Introduction to Numerical Analysis | publisher=[[Physics:Springer-Verlag|Springer-Verlag]] | location=Berlin, New York | edition=3rd | isbn=978-0-387-95452-3 | year=2002}}.</ref>{{rp|243}}
* The pseudoinverse of the pseudoinverse is the original matrix: <math>\left(A^+\right)^+ = A</math>.<ref name="SB2002" />{{rp|245}}
* Pseudoinversion commutes with transposition, complex conjugation, and taking the conjugate transpose:<ref name="SB2002" />{{rp|245}} <!-- reference only mentions the last bit --> <math display="block">\left(A^\operatorname{T}\right)^+ = \left(A^+\right)^\operatorname{T}, \quad \left(\overline{A}\right)^+ = \overline{A^+}, \quad \left(A^*\right)^+ = \left(A^+\right)^* .</math>
* The pseudoinverse of a scalar multiple of {{tmath| A }} is the reciprocal multiple of {{tmath| A^+ }}:<math display="block">\left(\alpha A\right)^+ = \alpha^{-1} A^+</math> for {{tmath| \alpha \neq 0 }}.
* Ker and image of the pseudoinverse coincide with those of the conjugate transpose: <math>\ker\left(A^+\right) = \ker\left(A^*\right)</math> and <math>\operatorname{ran}\left(A^+\right) = \operatorname{ran}\left(A^*\right)</math>.

====Identities====
The following identity formula can be used to cancel or expand certain subexpressions involving pseudoinverses:
<math display="block">
A = {}A{}A^*{}A^{+*}{} = {}A^{+*}{}A^*{}A.
</math>
Equivalently, substituting <math>A^+</math> for <math>A</math> gives
<math display="block">
A^+ ={}A^+{}A^{+*}{}A^*{} = {}A^*{}A^{+*}{}A^+,
</math>
while substituting <math>A^*</math> for <math>A</math> gives
<math display="block">
A^* ={}A^*{}A{}A^+{}={}A^+{}A{}A^*.
</math>

===Reduction to Hermitian case===
The computation of the pseudoinverse is reducible to its construction in the Hermitian case. This is possible through the equivalences:
<math display="block">A^+ = \left(A^*A\right)^+ A^*,</math>
<math display="block">A^+ = A^* \left(A A^*\right)^+,</math>

as {{tmath| A^*A }} and {{tmath| A A^* }} are Hermitian.

===Pseudoinverse of products===

The equality {{tmath|1= (AB)^+ = B^+ A^+ }} does not hold in general. Rather, suppose {{tmath| A \in \mathbb{K}^{m\times n},\ B \in \mathbb{K}^{n\times p} }}. Then the following are equivalent:<ref>{{Cite journal|last=Greville|first=T. N. E.|date=1966-10-01|title=Note on the Generalized Inverse of a Matrix Product|url=https://epubs.siam.org/doi/10.1137/1008107|journal=SIAM Review|volume=8|issue=4|pages=518–521|doi=10.1137/1008107|bibcode=1966SIAMR...8..518G |issn=0036-1445}}</ref>

# <math display="block">(AB)^+ = B^+ A^+</math>
# <math display="block">\begin{aligned}
A^+ A BB^* A^* & = BB^* A^*, \\
BB^+ A^* A B & = A^* A B.
\end{aligned}</math>
# <math display="block">\begin{aligned}
\left(A^+ A BB^*\right)^* & = A^+ A BB^*, \\
\left(A^* A BB^+\right)^* & = A^* A BB^+.
\end{aligned}</math>
# <math display="block">A^+ A BB^* A^* A BB^+ = BB^* A^* A</math>
# <math display="block">\begin{aligned}
 A^+ A B & = B (AB)^+ AB, \\
BB^+ A^* & = A^* A B (AB)^+.
\end{aligned}</math>

The following are sufficient conditions for {{tmath|1= (AB)^+ = B^+ A^+ }}:
# {{tmath| A }} has orthonormal columns (then <math>A^*A = A^+ A = I_n</math>), &nbsp; or
# {{tmath| B }} has orthonormal rows (then <math>BB^* = BB^+ = I_n</math>), &nbsp; or
# {{tmath| A }} has linearly independent columns (then <math>A^+ A = I</math> ) and {{tmath| B }} has linearly independent rows (then <math>BB^+ = I</math>), &nbsp; or
# <math>B = A^*</math>, or
# <math>B = A^+</math>.

The following is a necessary condition for {{tmath|1= (AB)^+ = B^+ A^+ }}:
# <math>(A^+ A) (BB^+) = (BB^+) (A^+ A)</math>

The fourth sufficient condition yields the equalities
<math display="block">\begin{align}
\left(A A^*\right)^+ &= A^{+*} A^+, \\
\left(A^* A\right)^+ &= A^+ A^{+*}.
\end{align}</math>

Here is a counterexample where {{tmath|1= (AB)^+ \neq B^+ A^+ }}:

<math display="block">\Biggl( \begin{pmatrix} 1 & 1 \\ 0 & 0 \end{pmatrix} \begin{pmatrix} 0 & 0 \\ 1 & 1 \end{pmatrix} \Biggr)^+ = \begin{pmatrix} 1 & 1 \\ 0 & 0 \end{pmatrix}^+ = \begin{pmatrix}
 \tfrac12 & 0 \\ \tfrac12 & 0 \end{pmatrix} \quad \neq \quad \begin{pmatrix}
 \tfrac14 & 0 \\ \tfrac14 & 0 \end{pmatrix} = \begin{pmatrix} 0 & \tfrac12 \\ 0 & \tfrac12 \end{pmatrix} \begin{pmatrix} \tfrac12 & 0 \\ \tfrac12 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 1 & 1 \end{pmatrix}^+ \begin{pmatrix} 1 & 1 \\ 0 & 0 \end{pmatrix}^+</math>

===Projectors===
<math>P = A A^+</math> and <math>Q = A^+A</math> are [[Projection (linear algebra)|orthogonal projection operators]], that is, they are Hermitian (<math>P = P^*</math>, <math>Q = Q^*</math>) and idempotent (<math>P^2 = P</math> and <math>Q^2 = Q</math>). The following hold:
* <math>PA = AQ = A</math> and <math>A^+ P = QA^+ = A^+</math>
* {{tmath| P }} is the orthogonal projector onto the [[Range of a function|range]] of {{tmath| A }} (which equals the [[Orthogonal complement|orthogonal complement]] of the kernel of {{tmath| A^* }}).
* {{tmath| Q }} is the orthogonal projector onto the range of {{tmath| A^* }} (which equals the orthogonal complement of the kernel of {{tmath| A }}).
* <math>I - Q = I - A^+A</math> is the orthogonal projector onto the kernel of {{tmath| A }}.
* <math>I - P = I - A A^+</math> is the orthogonal projector onto the kernel of {{tmath| A^* }}.<ref name="GvL1996"/>

The last two properties imply the following identities:
* <math>A\,\ \left(I - A^+ A\right)= \left(I - A A^+\right)A\ \ = 0</math>
* <math>A^*\left(I - A A^+\right) = \left(I - A^+A\right)A^* = 0</math>

Another property is the following: if {{tmath| A \in \mathbb{K}^{n\times n} }} is Hermitian and idempotent (true if and only if it represents an orthogonal projection), then, for any matrix {{tmath| B\in \mathbb{K}^{m\times n} }} the following equation holds:<ref>{{cite journal|first1=Anthony A.|last1=Maciejewski|first2=Charles A.|last2=Klein|title=Obstacle Avoidance for Kinematically Redundant Manipulators in Dynamically Varying Environments|journal=International Journal of Robotics Research|volume=4|issue=3|pages=109–117|year=1985|doi=10.1177/027836498500400308|hdl=10217/536|s2cid=17660144|hdl-access=free}}</ref>
<math display="block">A(BA)^+ = (BA)^+</math>

This can be proven by defining matrices <math>C = BA</math>, <math>D = A(BA)^+</math>, and checking that {{tmath| D }} is indeed a pseudoinverse for {{tmath| C }} by verifying that the defining properties of the pseudoinverse hold, when {{tmath| A }} is Hermitian and idempotent.

From the last property it follows that, if {{tmath| A \in \mathbb{K}^{n\times n} }} is Hermitian and idempotent, for any matrix {{tmath| B \in \mathbb{K}^{n\times m} }}
<math display="block">(AB)^+A = (AB)^+</math>

Finally, if {{tmath| A }} is an orthogonal projection matrix, then its pseudoinverse trivially coincides with the matrix itself, that is, <math>A^+ = A</math>.

===Geometric construction===
If we view the matrix as a linear map {{tmath| A:\mathbb{K}^n \to \mathbb{K}^m }} over the field {{tmath| \mathbb{K} }} then {{tmath| A^+: \mathbb{K}^m \to \mathbb{K}^n }} can be decomposed as follows. We write {{tmath| \oplus }} for the [[Direct sum of modules|direct sum]], {{tmath| \perp }} for the [[Orthogonal complement|orthogonal complement]], {{tmath| \ker }} for the [[Kernel (linear algebra)|kernel]] of a map, and {{tmath| \operatorname{ran} }} for the image of a map. Notice that <math>\mathbb{K}^n = \left(\ker A\right)^\perp \oplus \ker A</math> and <math>\mathbb{K}^m = \operatorname{ran} A \oplus \left(\operatorname{ran} A\right)^\perp</math>. The restriction <math> A: \left(\ker A\right)^\perp \to \operatorname{ran} A</math> is then an isomorphism. This implies that {{tmath| A^+ }} on {{tmath| \operatorname{ran} A }} is the inverse of this isomorphism, and is zero on <math>\left(\operatorname{ran} A\right)^\perp .</math>

In other words: To find {{tmath| A^+b }} for given {{tmath| b }} in {{tmath| \mathbb{K}^m }}, first project {{tmath| b }} orthogonally onto the range of {{tmath| A }}, finding a point {{tmath| p(b) }} in the range. Then form {{tmath| A^{-1}(\{p(b)\}) }}, that is, find those vectors in {{tmath| \mathbb{K}^n }} that {{tmath| A }} sends to {{tmath| p(b) }}. This will be an affine subspace of {{tmath| \mathbb{K}^n }} parallel to the kernel of {{tmath| A }}. The element of this subspace that has the smallest length (that is, is closest to the origin) is the answer {{tmath| A^+b }} we are looking for. It can be found by taking an arbitrary member of {{tmath| A^{-1}(\{p(b)\}) }} and projecting it orthogonally onto the orthogonal complement of the kernel of {{tmath| A }}.

This description is closely related to the [[Moore–Penrose inverse#Minimum norm solution to a linear system|minimum-norm solution to a linear system]].

===Limit relations===
The pseudoinverse are limits:
<math display="block">A^+ = \lim_{\delta \searrow 0} \left(A^* A + \delta I\right)^{-1} A^*
= \lim_{\delta \searrow 0} A^* \left(A A^* + \delta I\right)^{-1}
</math>
(see [[Tikhonov regularization]]). These limits exist even if {{tmath| \left(A A^*\right)^{-1} }} or {{tmath| \left(A^*A\right)^{-1} }} do not exist.<ref name="GvL1996"/>{{rp|263}}

===Continuity===
In contrast to ordinary matrix inversion, the process of taking pseudoinverses is not [[Continuous function|continuous]]: if the sequence {{tmath| \left(A_n\right) }} converges to the matrix {{tmath| A }} (in the [[Matrix norm|maximum norm or Frobenius norm]], say), then {{tmath| (A_n)^+ }} need not converge to {{tmath| A^+ }}. However, if all the matrices {{tmath| A_n}} have the same rank as {{tmath| A }}, {{tmath| (A_n)^+ }} will converge to {{tmath| A^+ }}.<ref name="rakocevic1997">{{cite journal | last=Rakočević | first=Vladimir | title=On continuity of the Moore–Penrose and Drazin inverses | journal=Matematički Vesnik | volume=49 | pages=163–72 | year=1997 | url =http://elib.mi.sanu.ac.rs/files/journals/mv/209/mv973404.pdf }}</ref>

===Derivative===
The derivative of a real-valued pseudoinverse matrix that has constant rank at a point {{tmath| x }} may be calculated in terms of the derivative of the original matrix:<ref>{{cite journal|title=The Differentiation of Pseudo-Inverses and Nonlinear Least Squares Problems Whose Variables Separate|first1=G. H.|last1=Golub|first2=V.|last2=Pereyra|journal=SIAM Journal on Numerical Analysis|volume=10|number=2|date=April 1973|pages=413–32|jstor=2156365|doi=10.1137/0710036|bibcode=1973SJNA...10..413G}}</ref>
<math display="block">
\frac{\mathrm d}{\mathrm d x} A^+(x) =
-A^+ \left( \frac{\mathrm d}{\mathrm d x} A \right) A^+ ~+~
A^+ A^{+\textsf{T}} \left(\frac{\mathrm d}{\mathrm d x} A^\operatorname{T}\right) \left(I - A A^+\right) ~+~
\left(I - A^+ A\right) \left(\frac{\text{d}}{\text{d} x} A^\operatorname{T}\right) A^{+\textsf{T}} A^+.
</math>
For a complex matrix, the transpose is replaced with conjugate transpose.<ref>{{cite book |last1=Hjørungnes |first1=Are |title=Complex-valued matrix derivatives: with applications in signal processing and communications |date=2011 |publisher=Cambridge university press |location=New York |isbn=9780521192644 |page=52}}</ref> For a real-valued symmetric matrix, the Magnus-Neudecker derivative is established.<ref>{{Cite journal| last1=Liu|first1=Shuangzhe|
last2= Trenkler|first2=Götz| last3=Kollo|first3=Tõnu| 
last4=von Rosen|first4=Dietrich| 
last5=Baksalary|first5=Oskar Maria| 
date= 2023|
title= Professor Heinz Neudecker and matrix differential calculus| 
journal= Statistical Papers| language=en|
doi= 10.1007/s00362-023-01499-w}}</ref>

==Examples==
Since for invertible matrices the pseudoinverse equals the usual inverse, only examples of non-invertible matrices are considered below.

* For <math>A = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix},</math> the pseudoinverse is <math>A^+ = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}.</math> The uniqueness of this pseudoinverse can be seen from the requirement <math>A^+ = A^+ A A^+</math>, since multiplication by a zero matrix would always produce a zero matrix.
* For <math>A = \begin{pmatrix} 1 & 0 \\ 1 & 0 \end{pmatrix},</math> the pseudoinverse is <math>A^+ = \begin{pmatrix} \frac{1}{2} & \frac{1}{2} \\ 0 & 0 \end{pmatrix}</math>.
: Indeed, <math>A\,A^+ = \begin{pmatrix} \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} \end{pmatrix}</math>, and thus <math>A\,A^+ A = \begin{pmatrix} 1 & 0 \\ 1 & 0\end{pmatrix} = A</math>. Similarly, <math>A^+A = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}</math>, and thus <math>A^+A\,A^+ = \begin{pmatrix} \frac{1}{2} & \frac{1}{2} \\ 0 & 0 \end{pmatrix} = A^+</math>.
: Note that {{tmath| A }} is neither injective nor surjective, and thus the pseudoinverse cannot be computed via <math>A^+ = \left(A^* A\right)^{-1} A^*</math> nor <math>A^+ = A^* \left( A A^*\right)^{-1}</math>, as <math>A^* A</math> and <math>A A^*</math> are both singular, and furthermore <math>A^+</math> is neither a left nor a right inverse.
: Nonetheless, the pseudoinverse can be computed via SVD observing that <math>A=\sqrt2 \left(\frac{\mathbf e_1+\mathbf e_2}{\sqrt2}\right) \mathbf e_1^*</math>, and thus <math>A^+=\frac{1}{\sqrt2} \,\mathbf e_1 \left(\frac{\mathbf e_1+\mathbf e_2}{\sqrt2}\right)^*</math>.
* For <math>A = \begin{pmatrix} 1 & 0 \\ -1 & 0 \end{pmatrix},</math> <math>A^+ = \begin{pmatrix} \frac{1}{2} & -\frac{1}{2} \\ 0 & 0 \end{pmatrix}.</math>
* For <math>A = \begin{pmatrix} 1 & 0 \\ 2 & 0 \end{pmatrix},</math> <math>A^+ = \begin{pmatrix} \frac{1}{5} & \frac{2}{5} \\ 0 & 0 \end{pmatrix}</math>. The denominators are here <math>5 = 1^2 + 2^2</math>.
* For <math>A = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix},</math> <math>A^+ = \begin{pmatrix} \frac{1}{4} & \frac{1}{4} \\ \frac{1}{4} & \frac{1}{4} \end{pmatrix}.</math>
* For <math>A = \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ 0 & 1 \end{pmatrix},</math> the pseudoinverse is <math>A^+ = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \frac{1}{2} & \frac{1}{2} \end{pmatrix}</math>.
: For this matrix, the [[Inverse element#Matrices|left inverse]] exists and thus equals <math>A^+</math>, indeed, <math>A^+A = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}.</math>


==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 {{tmath| x }} is zero if {{tmath| x }} is zero and the reciprocal of {{tmath| x }} otherwise:
<math display="block">x^+ = \begin{cases}
0, & \mbox{if }x = 0; \\ 
x^{-1}, & \mbox{otherwise}.
\end{cases}</math>

===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:

=== <math display="block">\vec{x}^+ = \begin{cases}
\vec{0}^\operatorname{T}, & \text{if } \vec{x} = \vec{0}; \\
\dfrac{\vec{x}^*}{\vec{x}^* \vec{x}}, & \text{otherwise}.
\end{cases}</math>Diagonal matrices ===
The pseudoinverse of a squared diagonal matrix is obtained by taking the reciprocal of the nonzero diagonal elements. Formally, if <math>D</math> is a squared diagonal matrix with <math>D=\tilde D\oplus \mathbf 0_{k\times k}</math> and <math>\tilde D>0</math>, then <math>D^+=\tilde D^{-1}\oplus \mathbf 0_{k\times k}</math>. More generally, if <math>A</math> is any <math>m\times n</math> rectangular matrix whose only nonzero elements are on the diagonal, meaning <math>A_{ij}=\delta_{ij} a_i</math>, <math>a_i\in\mathbb K</math>, then <math>A^+</math> is a <math>n\times m</math> rectangular matrix whose diagonal elements are the reciprocal of the original ones, that is, <math>A_{ii}\neq 0\implies A^+_{ii}=1/A_{ii}</math>.

===Linearly independent columns===
If the '''rank''' of {{tmath| A }} is identical to its '''column rank''', {{tmath| n }}, (for {{tmath| n \le m}},) there are {{tmath| n }} [[Linear independence|linearly independent]] columns, and {{tmath| A^*A }} is invertible. In this case, an explicit formula is:{{sfn|Ben-Israel|Greville|2003}}
<math display="block">A^+ = \left(A^*A\right)^{-1}A^*.</math>

It follows that {{tmath| A^+ }} is then a left inverse of {{tmath| A }}: &nbsp; <math>A^+ A = I_n</math>.

===Linearly independent rows===
If the '''rank''' of {{tmath| A }} is identical to its '''row rank''', {{tmath| m }}, (for {{tmath| m \le n}},) there are {{tmath| m }} [[Linear independence|linearly independent]] rows, and {{tmath| AA^* }} is invertible. In this case, an explicit formula is:
<math display="block">A^+ = A^*\left(A A^*\right)^{-1}.</math>

It follows that {{tmath| A^+ }} is a right inverse of {{tmath| A }}: &nbsp; <math>A A^+ = I_m</math>.

===Orthonormal columns or rows===
This is a special case of either full column rank or full row rank (treated above). If {{tmath| A }} has orthonormal columns (<math>A^*A = I_n</math>) or orthonormal rows (<math>A A^* = I_m</math>), then:
<math display="block">A^+ = A^* .</math>

=== Normal matrices ===
If {{tmath| A }} is [[Normal matrix|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 {{tmath| A }} commuting with its transpose implies that it commutes with its pseudoinverse.

===Orthogonal projection matrices===
This is a special case of a normal matrix with eigenvalues 0 and 1. If {{tmath| A }} is an orthogonal projection matrix, that is, <math>A = A^*</math> and <math>A^2 = A</math>, then the pseudoinverse trivially coincides with the matrix itself:
<math display="block">A^+ = A.</math>

===Circulant matrices===
For a [[Circulant matrix|circulant matrix]] {{tmath| C }}, the singular value decomposition is given by the [[Fourier transform]], that is, the singular values are the Fourier coefficients. Let {{tmath| \mathcal{F} }} be the [[DFT matrix|Discrete Fourier Transform (DFT) matrix]]; then<ref name="Stallings1972">{{cite journal | last1=Stallings | first1=W. T. |  title=The Pseudoinverse of an ''r''-Circulant Matrix | journal=[[Organization:Proceedings of the American Mathematical Society|Proceedings of the American Mathematical Society]] | volume=34 | issue=2 | pages=385–88 | year=1972 | doi=10.2307/2038377 | last2=Boullion | first2=T. L.| jstor=2038377 }}</ref>
<math display="block">\begin{align}
C &= \mathcal{F}\cdot\Sigma\cdot\mathcal{F}^*,\\
C^+ &= \mathcal{F}\cdot\Sigma^+\cdot\mathcal{F}^*.
\end{align}</math>

==Construction==
===Rank decomposition===
Let {{tmath| r \le \min(m, n) }} denote the rank of {{tmath| A \in \mathbb{K}^{m\times n} }}. Then {{tmath| A }} can be [[Rank factorization|(rank) decomposed]] as
<math>A = BC</math> where {{tmath| B \in \mathbb{K}^{m\times r} }} and {{tmath| C \in \mathbb{K}^{r\times n} }} are of rank {{tmath| r }}. Then <math>A^+ = C^+B^+ = C^*\left(CC^*\right)^{-1}\left(B^*B\right)^{-1}B^*</math>.

===The QR method===
For <math>\mathbb{K} \in \{ \mathbb{R}, \mathbb{C}\}</math> computing the product {{tmath| A A^* }} or {{tmath| A^*A }} 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 {{tmath| A }} may be used instead.

Consider the case when {{tmath| A }} is of full column rank, so that <math>A^+ = \left(A^*A\right)^{-1}A^*</math>. Then the [[Cholesky decomposition]] <math>A^*A = R^*R</math>, where {{tmath| R }} 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,
<math display="block">A^+ = \left(A^*A\right)^{-1}A^* \quad \Leftrightarrow \quad \left(A^*A\right)A^+ = A^* \quad \Leftrightarrow \quad R^*RA^+ = A^*</math>

which may be solved by forward substitution followed by back substitution.

The Cholesky decomposition may be computed without forming {{tmath| A^*A }} explicitly, by alternatively using the [[QR decomposition]] of <math>A = Q R</math>, where <math>Q</math> has orthonormal columns, <math>Q^*Q = I</math>, and {{tmath| R }} is upper triangular. Then
<math display="block">A^*A \,=\, (Q R)^*(Q R) \,=\, R^*Q^*Q R \,=\, R^*R ,</math>

so {{tmath| R }} is the Cholesky factor of {{tmath| A^*A }}.

The case of full row rank is treated similarly by using the formula <math>A^+ = A^*\left(A A^*\right)^{-1}</math> and using a similar argument, swapping the roles of {{tmath| A }} and {{tmath| A^* }}.

===Singular value decomposition (SVD)===
A computationally simple and accurate way to compute the pseudoinverse is by using the [[Singular value decomposition|singular value decomposition]].{{sfn|Ben-Israel|Greville|2003}}<ref name="GvL1996"/><ref name="SLEandPI">[http://websites.uwlax.edu/twill/svd/systems/index.html Linear Systems & Pseudo-Inverse]</ref> If <math>A = U\Sigma V^*</math> is the singular value decomposition of {{tmath| A }}, then <math>A^+ = V\Sigma^+ U^*</math>. For a rectangular diagonal matrix such as {{tmath| \Sigma }}, 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 [[Software:MATLAB|MATLAB]] or [[Software:GNU Octave|GNU Octave]] function {{mono|pinv}}, the tolerance is taken to be {{math|1=''t'' = ε⋅max(''m'', ''n'')⋅max(Σ)}}, where ε is the [[Machine epsilon|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 [[Software:LAPACK|LAPACK]]) is used.

The above procedure shows why taking the pseudoinverse is not a continuous operation: if the original matrix {{tmath| A }} has a singular value 0 (a diagonal entry of the matrix {{tmath| \Sigma }} above), then modifying {{tmath| A }} 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===
[[Block matrix pseudoinverse|Optimized approaches]] exist for calculating the pseudoinverse of block-structured matrices.

===The iterative method of Ben-Israel and Cohen===
Another method for computing the pseudoinverse (cf. [[Drazin inverse]]) uses the recursion
<math display="block">A_{i+1} = 2A_i - A_i A A_i,</math>

which is sometimes referred to as hyper-power sequence. This recursion produces a sequence converging quadratically to the pseudoinverse of {{tmath| A }} if it is started with an appropriate {{tmath| A_0 }} satisfying <math>A_0 A = \left(A_0 A\right)^*</math>. The choice <math>A_0 = \alpha A^*</math> (where <math>0 < \alpha < 2/\sigma^2_1(A)</math>, with {{tmath| \sigma_1(A) }} denoting the largest singular value of {{tmath| A }})<ref>{{cite journal | last1=Ben-Israel | first1=Adi | last2=Cohen | first2=Dan | title=On Iterative Computation of Generalized Inverses and Associated Projections | journal=SIAM Journal on Numerical Analysis | volume=3 | issue=3 | pages=410–19 | year=1966 | jstor=2949637 | doi=10.1137/0703035 | bibcode=1966SJNA....3..410B }}[http://benisrael.net/COHEN-BI-ITER-GI.pdf pdf]</ref> 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 {{tmath| A_i }} enters the region of quadratic convergence.<ref>{{cite journal | last1=Söderström | first1=Torsten | last2=Stewart | first2=G. W. | title=On the Numerical Properties of an Iterative Method for Computing the Moore–Penrose Generalized Inverse | journal=SIAM Journal on Numerical Analysis | volume=11 | issue=1 | pages=61–74 | year=1974 | jstor=2156431 | doi=10.1137/0711008 | bibcode=1974SJNA...11...61S }}</ref> However, if started with {{tmath| A_0 }} already close to the Moore–Penrose inverse and <math>A_0 A = \left(A_0 A\right)^*</math>, for example <math>A_0 := \left(A^* A + \delta I\right)^{-1} A^*</math>, convergence is fast (quadratic).

===Updating the pseudoinverse===
For the cases where {{tmath| A }} has full row or column rank, and the inverse of the correlation matrix ({{tmath| A A^* }} for {{tmath| A }} with full row rank or {{tmath| A^*A }} for full column rank) is already known, the pseudoinverse for matrices related to {{tmath| A }} 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.<ref name="G1992">{{Cite thesis |first= Tino |last=Gramß |title= Worterkennung mit einem künstlichen neuronalen Netzwerk |type=PhD dissertation |publisher= Georg-August-Universität zu Göttingen |year = 1992 | oclc = 841706164 }}</ref><ref name="EMTIYAZ2008">{{cite web |first=Mohammad |last=Emtiyaz |title=Updating Inverse of a Matrix When a Column is Added/Removed |date=February 27, 2008 |url=https://emtiyaz.github.io/Writings/OneColInv.pdf }}</ref>

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.<ref>{{cite journal|last=Meyer|first=Carl D. Jr.|title=Generalized inverses and ranks of block matrices|journal=SIAM J. Appl. Math.|volume=25|issue=4|date=1973|pages=597–602|doi=10.1137/0125057}}</ref><ref>{{cite journal|last=Meyer|first=Carl D. Jr.|title=Generalized inversion of modified matrices|journal=SIAM J. Appl. Math.|volume=24|issue=3|date=1973|pages=315–23|doi=10.1137/0124033}}</ref>

===Software libraries===
High-quality implementations of SVD, QR, and back substitution are available in standard libraries, such as [[Software:LAPACK|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|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 [[Software:NumPy|NumPy]] provides a pseudoinverse calculation through its functions <code>matrix.I</code> and <code>linalg.pinv</code>; its <code>pinv</code> uses the SVD-based algorithm. [[Software:SciPy|SciPy]] adds a function <code>scipy.linalg.pinv</code> that uses a least-squares solver. 

The MASS package for [[R (programming language)|R]] provides a calculation of the Moore–Penrose inverse through the <code>ginv</code> function.<ref>{{cite web |url=https://stat.ethz.ch/R-manual/R-devel/library/MASS/html/ginv.html |title=R: Generalized Inverse of a Matrix}}</ref> The <code>ginv</code> function calculates a pseudoinverse using the singular value decomposition provided by the <code>svd</code> function in the base R package. An alternative is to employ the <code>pinv</code> function available in the pracma package.

The [[Software:GNU Octave|Octave programming language]] provides a pseudoinverse through the standard package function <code>pinv</code> and the <code>pseudo_inverse()</code> method.

In [[Julia (programming language)]], the LinearAlgebra package of the standard library provides an implementation of the Moore–Penrose inverse <code>pinv()</code> implemented via singular-value decomposition.<ref>{{cite web |url=https://docs.julialang.org/en/v1/stdlib/LinearAlgebra/#LinearAlgebra.pinv |title=LinearAlgebra.pinv}}</ref>

==Applications==
===Linear least-squares===

The pseudoinverse provides a least squares solution to a [[System of linear equations|system of linear equations]].<ref name="Penrose1956">{{cite journal | last=Penrose | first=Roger | title=On best approximate solution of linear matrix equations | journal=Proceedings of the Cambridge Philosophical Society | volume=52 | pages=17–19 | year=1956 | issue=1 | doi=10.1017/S0305004100030929| bibcode=1956PCPS...52...17P | s2cid=122260851 }}</ref>
For {{tmath| A \in \mathbb{K}^{m\times n} }}, given a system of linear equations
<math display="block">A x = b,</math>

in general, a vector {{tmath| x }} 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 <math>b</math> is in the image of <math>A</math>, and is unique if and only if <math>A</math> is injective. The pseudoinverse solves the "least-squares" problem as follows:

* {{tmath| \forall x \in \mathbb{K}^n }}, we have <math>\left\|Ax - b\right\|_2 \ge \left\|Az - b\right\|_2</math> where <math>z = A^+b</math> and <math>\|\cdot\|_2</math> denotes the Euclidean norm. This weak inequality holds with equality if and only if <math>x = A^+b + \left(I - A^+A\right)w</math> for any vector {{tmath| w }}; this provides an infinitude of minimizing solutions unless {{tmath| A }} has full column rank, in which case {{tmath| \left(I - A^+A\right) }} is a zero matrix.<ref name=Planitz>{{cite journal|last=Planitz|first=M.|title=Inconsistent systems of linear equations|journal=Mathematical Gazette|volume=63|issue=425|date=October 1979|pages=181–85|doi=10.2307/3617890|jstor=3617890|s2cid=125601192 }}</ref> The solution with minimum Euclidean norm is {{tmath| z. }}<ref name=Planitz/>

This result is easily extended to systems with multiple right-hand sides, when the Euclidean norm is replaced by the Frobenius norm. Let {{tmath| B \in \mathbb{K}^{m\times p} }}.

* {{tmath| \forall X \in \mathbb{K}^{n\times p} }}, we have <math>\|AX - B\|_{\mathrm{F}} \ge \|AZ -B\|_{\mathrm{F}}</math> where <math>Z = A^+B</math> and <math>\|\cdot\|_{\mathrm{F}}</math> denotes the Frobenius norm.

===Obtaining all solutions of a linear system===
If the linear system

<math display="block">A x = b</math>

has any solutions, they are all given by<ref name=James>{{cite journal|last=James|first=M.|title=The generalised inverse|journal=Mathematical Gazette|volume=62|issue=420|date=June 1978|pages=109–14|doi=10.1017/S0025557200086460|s2cid=126385532 }}</ref>

<math display="block">x = A^+ b + \left[I - A^+ A\right]w</math>

for arbitrary vector {{tmath| w }}. Solution(s) exist if and only if <math>A A^+ b = b</math>.<ref name=James/> If the latter holds, then the solution is unique if and only if {{tmath| A }} has full column rank, in which case {{tmath| I - A^+ A }} is a zero matrix. If solutions exist but {{tmath| A }} does not have full column rank, then we have an [[Indeterminate system|indeterminate system]], all of whose infinitude of solutions are given by this last equation.

===Minimum norm solution to a linear system===
For linear systems <math>Ax = b,</math> with non-unique solutions (such as under-determined systems), the pseudoinverse may be used to construct the solution of minimum Euclidean norm
<math>\|x\|_2</math> among all solutions.

* If <math>Ax = b</math> is satisfiable, the vector <math>z = A^+b</math> is a solution, and satisfies <math>\|z\|_2 \le \|x\|_2</math> 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 {{tmath| B \in \mathbb{K}^{m\times p} }}.

* If <math>AX = B</math> is satisfiable, the matrix <math>Z = A^+B</math> is a solution, and satisfies <math>\|Z\|_{\mathrm{F}} \le \|X\|_{\mathrm{F}}</math> for all solutions.

===Condition number===
Using the pseudoinverse and a [[Matrix norm|matrix norm]], one can define a [[Condition number|condition number]] for any matrix:
<math display="block">\mbox{cond}(A) = \|A\| \left\|A^+\right\|.</math>

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 {{tmath| A }} can lead to huge errors in the entries of the solution.<ref name=hagen/>

==Generalizations==
In order to solve more general least-squares problems, one can define Moore–Penrose inverses for all continuous linear operators {{tmath| A: H_1 \rarr H_2 }} between two [[Hilbert space]]s {{tmath| H_1 }} and {{tmath| H_2 }}, 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.<ref name="hagen">{{cite book|first1=Roland|last1=Hagen|first2=Steffen|last2=Roch|first3=Bernd|last3=Silbermann|title=C*-algebras and Numerical Analysis|publisher=CRC Press|year=2001|chapter=Section 2.1.2}}</ref> Those that do are precisely the ones whose range is [[Closed set|closed]] in {{tmath| H_2 }}.

A notion of pseudoinverse exists for matrices over an arbitrary [[Field (mathematics)|field]] equipped with an arbitrary [[Involution (mathematics)|involutive]] [[Automorphism|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 <math>\operatorname{rank}(A) = \operatorname{rank}\left(A^* A\right) = \operatorname{rank}\left(A A^*\right)</math>, where <math>A^*</math> denotes the result of applying the involution operation to the transpose of <math>A</math>. When it does exist, it is unique.<ref>{{Cite journal|last=Pearl|first=Martin H.|date=1968-10-01|title=Generalized inverses of matrices with entries taken from an arbitrary field|journal=Linear Algebra and Its Applications|language=en|volume=1|issue=4|pages=571–587|doi=10.1016/0024-3795(68)90028-1|issn=0024-3795|doi-access=free}}</ref> '''Example''': Consider the field of complex numbers equipped with the [[Identity function|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 <math>A = \begin{bmatrix}1 & i\end{bmatrix}^\operatorname{T}</math>. Observe that <math>\operatorname{rank}\left(A A^\operatorname{T}\right) = 1</math> while <math>\operatorname{rank}\left(A^\operatorname{T} A\right) = 0</math>. So this matrix doesn't have a pseudoinverse in this sense.

In [[Abstract algebra|abstract algebra]], a Moore–Penrose inverse may be defined on a [[*-regular semigroup|*-regular semigroup]]. This abstract definition coincides with the one in linear algebra.

==See also==
* [[Drazin inverse]]
* Hat matrix
* [[Inverse element]]
* Linear least squares (mathematics)
* [[Pseudo-determinant]]
* [[Von Neumann regular ring]]

==Notes==
{{Reflist|30em}}

==References==
* {{cite book| first1 = Adi | last1 = Ben-Israel |author1-link=Adi Ben-Israel| first2 = Thomas N.E. | last2 = Greville |author2-link=Thomas N. E. Greville|title=Generalized inverses: Theory and applications|
edition= 2nd |location= New York, NY | publisher = Springer |year=2003| isbn = 978-0-387-00293-4 | doi = 10.1007/b97366 }}
* {{cite book| first1 = S. L. | last1 = Campbell | first2 = C. D. Jr.| last2 = Meyer | title= Generalized Inverses of Linear Transformations| url = https://archive.org/details/generalizedinver0000camp | url-access = registration |
publisher=Dover |year=1991 | isbn = 978-0-486-66693-8 }}
* {{cite book| first = Yoshihiko | last = Nakamura | title= Advanced Robotics: Redundancy and Optimization|
publisher=Addison-Wesley |year= 1991 | isbn = 978-0201151985 }}
* {{cite book| first1 = C. Radhakrishna | last1 = Rao | first2 = Sujit Kumar | last2 = Mitra| title = Generalized Inverse of Matrices and its Applications | url = https://archive.org/details/generalizedinver0000raoc | url-access = registration | publisher= John Wiley & Sons |location = New York |year= 1971 | page=[https://archive.org/details/generalizedinver0000raoc/page/240 240] | isbn = 978-0-471-70821-6 }}

==External links==
* {{PlanetMath |urlname=Pseudoinverse |title=Pseudoinverse}}
* [http://people.revoledu.com/kardi/tutorial/LinearAlgebra/MatrixGeneralizedInverse.html Interactive program & tutorial of Moore–Penrose Pseudoinverse]
* {{PlanetMath |urlname=MoorePenroseGeneralizedInverse |title=Moore–Penrose generalized inverse}}
* {{MathWorld|urlname=Pseudoinverse|title=Pseudoinverse}}
* {{MathWorld|urlname=Moore-PenroseMatrixInverse|title=Moore–Penrose Inverse}}
* [https://arxiv.org/abs/1110.6882 The Moore–Penrose Pseudoinverse. A Tutorial Review of the Theory]
* [http://engineerjs.com/doc/ejs/engine/linalg-1/_pinv.html Online Moore–Penrose Inverse calculator]

{{Numerical linear algebra}}
{{Roger Penrose}}

{{DEFAULTSORT:Moore-Penrose inverse}}
[[Category:Matrix theory]]
[[Category:Singular value decomposition]]
[[Category:Numerical linear algebra]]

{{Sourceattribution|Moore–Penrose inverse}}