= Moore-Penrose inverse
{c}
{title2=$A^+$}
{wiki}
= Moore-Penrose pseudoinverse
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{synonym}
= Pseudoinverse
{synonym}
The <Moore-Penrose inverse> extends the useful part of a <matrix inverse> to rectangular and singular <matrices>. A <linear system> $Ax=b$ may have no exact solution, or may have many. The <pseudoinverse> resolves both ambiguities: $x_*=A^+b$ is the <minimum-norm least-squares solution>. First minimize the <Euclidean norm> $\|Ax-b\|_2$; among all <vectors> attaining that minimum, choose the one with smallest <Euclidean norm> $\|x\|_2$. It does not pretend that inconsistent equations can all be satisfied.
For $A\in\mathbb C^{m\times n}$, its <Moore-Penrose inverse> is the unique $n\times m$ <matrix> satisfying the four <Penrose equations>. The <singular value decomposition> gives a direct construction:
$$
A=U\Sigma V^*,
\qquad
\boxed{A^+=V\Sigma^+U^*.}
$$
Here $U,V$ are <unitary matrices>, the star is the <conjugate transpose>, and $\Sigma^+$ is the $n\times m$ rectangular <diagonal matrix> obtained by replacing each nonzero <singular value> $\sigma_j$ by $1/\sigma_j$ and leaving zeros at zero. Over the <real numbers>, use <orthogonal matrices> and <transposes>. The <pseudoinverse> reverses the stretches that can be reversed; it discards output components the original <linear map> cannot produce and does not invent an input component in its <kernel>. https://ocw.mit.edu/courses/18-06sc-linear-algebra-fall-2011/resources/mit18_06scf11_ses3-8sum/[Strang, MIT 18.06SC: Left and Right Inverses; Pseudoinverse]{ref}
The two products have especially clear meanings:
$$
AA^+=P_{\operatorname{ran}A},
\qquad
A^+A=P_{(\ker A)^\perp}.
$$
They are <orthogonal projection matrices>: $AA^+$ selects the producible part of an output, while $A^+A$ selects the detectable part of an input. They are generally not <identity matrices>. For an <invertible matrix>, both are identities and $A^+=A^{-1}$.
This gives a useful order of operations for <least-squares solutions>: project the requested output onto the <column space>, then invert the restriction of $A$ from $(\ker A)^\perp$ to its <column space>. Adding a <kernel> <vector> leaves the residual unchanged but increases the squared <Euclidean norm> by the <Pythagorean theorem in an inner-product space>, because $A^+b$ is <orthogonal> to that <kernel>. The <pseudoinverse> therefore provides one canonical answer without hiding either lost information or irreconcilable data.
The name records a history with several contributors. E. H. Moore gave an earlier generalization in 1920; M. H. Pearl's 1966 paper records that work, Arne Bjerhammar's 1951 rediscovery for maximal-rank rectangular <matrices>, and Roger Penrose's independent 1955 treatment. https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/abs/on-generalized-inverses-of-matrices/9CD637F604C76ECD8D03AC029212D56E[Pearl, On generalized inverses of matrices (1966), introduction and references 6–9]{ref} Penrose's original paper proves existence and uniqueness from the four <Penrose equations> for arbitrary complex <matrices>, including rectangular ones. https://faculty.kfupm.edu.sa/math/jaafarm/lec-notes/Moore-Pinrose.pdf[Penrose, A generalized inverse for matrices (1955), Theorem 1, pp. 406–408]{ref}
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