Moore-Penrose pseudoinverse

Definition

Consider $A \in \mathbb{R}_r^{m \times m}$, then $A$ has a unique pseudoinverse $A^+ \in \mathbb{R}_r^{m \times m}$.

Theorem (Penrose, 1956)

For $A \in \mathbb{R}_r^{m \times m}$, $G = A^+$ if and only if the following hold:

1. $AGA = A$
2. $GAG = G$
3. $(AG)^T = AG$
4. $(GA)^T = GA$

Furthermore $A^+$ always exists and is unique.

Theorem (Albert, 1972)

Let $A \in \mathbb{R}_r^{m \times m}$, then,

Notes

• This can be considered a generalization of the (two-sided) inverse of a matrix where $AA^{-1} = I = A^{-1}A$
  • compare also left inverse and right inverse
• may use SVD to compute it, e.g. for matrix $A$ if $A = U \Sigma V^T$, then $A^+ = V \Sigma^+ U^T$

References

1. https://www.math.ucla.edu/~laub/33a.2.12s/mppseudoinverse.pdf
2. G. Strang, Linear algebra and its applications, 4th ed., 2011, p. 122.
3. https://math.stackexchange.com/questions/458404/how-can-we-compute-pseudoinverse-for-any-matrix
4. https://ocw.mit.edu/courses/18-06sc-linear-algebra-fall-2011/0550c89b69c99e97dcbf52074e293308_MIT18_06SCF11_Ses3.8sum.pdf