The Pseudoinverse of $A=CR$ is $A^+=R^+C^+$ (?)

Abstract

This paper gives three formulas for the pseudoinverse of a matrix product $A = CR$. The first is sometimes correct, the second is always correct, and the third is almost never correct. But that third randomized pseudoinverse $A^+_r$ may be very useful when $A$ is a very large matrix. 1. $A^+ = R^+C^+$ when $A = CR$ and $C$ has independent columns and $R$ has independent rows. 2. $A^+ = (C^+CR)^+(CRR^+)^+$ is always correct. 3. $A^+_r = (P^TCR)^+P^TCRQ(CRQ)^+ = A^+$ only when $\mathrm{rank}(P^TA) = \mathrm{rank}(AQ) = \mathrm{rank}(A)$ with $A = CR$.