The Pseudoinverse of $A=CR$ is $A^+=R^+C^+$ (?)
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866917623083565056 |
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| author | Karpowicz, Michał P. Strang, Gilbert |
| author_facet | Karpowicz, Michał P. Strang, Gilbert |
| contents | 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 $A = CR$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_01716 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Pseudoinverse of $A=CR$ is $A^+=R^+C^+$ (?) Karpowicz, Michał P. Strang, Gilbert Numerical Analysis 15A09, 15A23, 15A24, 65F45, 65F55, 68W20 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 $A = CR$. |
| title | The Pseudoinverse of $A=CR$ is $A^+=R^+C^+$ (?) |
| topic | Numerical Analysis 15A09, 15A23, 15A24, 65F45, 65F55, 68W20 |
| url | https://arxiv.org/abs/2305.01716 |