References & Citations
Mathematics > Numerical Analysis
Title: The Pseudoinverse of $A=CR$ is $A^+=R^+C^+$ (?)
(Submitted on 2 May 2023 (v1), last revised 26 Mar 2024 (this version, v3))
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 $A = CR$.
Submission history
From: Michal Karpowicz Dr [view email] [via ASHLEY proxy][v1] Tue, 2 May 2023 18:28:22 GMT (5kb)
[v2] Sat, 6 May 2023 16:54:16 GMT (5kb)
[v3] Tue, 26 Mar 2024 22:23:16 GMT (136kb,D)
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