Moore–Penrose inverse

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.

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.