---
tag: 0PSRCMCM5X
kind: definition
name: Moore–Penrose inverse
msc: [15A09]
defines: 0YY8VX28JR
uses: [0EX045E2BP]
sources:
  - url: https://en.wikipedia.org/wiki/Moore–Penrose_inverse
    locator: Definition
---

## Definition

Let $A$ be an $m \times n$ complex matrix.
There is exactly one $n \times m$ complex matrix $A^+$ with these four properties:

1. $A A^+ A = A$.
2. $A^+ A A^+ = A^+$.
3. $(A A^+)^* = A A^+$.
4. $(A^+ A)^* = A^+ A$.

This matrix is the *Moore–Penrose inverse*, or *pseudoinverse*, of $A$, where $^*$ is the [[conjugate transpose|0DXKE10PHF]].

## Remarks

If $A$ is invertible, then $A^+ = A^{-1}$.
If the columns of $A$ are linearly independent, then $A^+ = (A^* A)^{-1} A^*$.
For $b \in \C^m$, the vector $A^+ b$ is the least squares solution of $Ax = b$ with the least norm.
If $A = U \Sigma V^*$ is a singular value decomposition, then $A^+ = V \Sigma^+ U^*$, where $\Sigma^+$ inverts the nonzero singular values and transposes.

## Explanations

### L1 human

The pseudoinverse extends the inverse to all matrices, also singular and rectangular ones.
It gives the best possible solution of $Ax = b$ when no exact solution exists or when there are many.
For the zero matrix, the pseudoinverse is the zero matrix of the transposed size.