The Pseudoinverse of $A=CR$ is $A^+=R^+C^+$ (?)
| Autor: | Karpowicz, Michał P., Strang, Gilbert |
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| Rok vydání: | 2023 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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 $A = CR$. Comment: 10 pages, 5 figures, matlab code, new paragraphs introduce general formulas for the pseudoinverse of CR, new Figures and the randomized pseudoinverse algorithm |
| Databáze: | arXiv |
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