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. $A^{+}=(C^{+}CR{)}^{+}(CR{R}^{+}{)}^{+}$ is always correct. 3. $A_r^{+}=(P^{T}CR{)}^{+}{P}^{T}CRQ(CRQ{)}^{+}=A^{+}$ only when $\mathrm{rank}(P^{T}A)=\mathrm{rank}(AQ)=\mathrm{rank}(A)$ with $A=CR$.

10 pages, 5 figures, matlab code, new paragraphs introduce general formulas for the pseudoinverse of CR, new Figures and the randomized pseudoinverse algorithm