Timeline for Additivity of purity of random matrix products
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Jul 22, 2023 at 17:53 | comment | added | Yaroslav Bulatov | @IosifPinelis OK this seems to be because for $X=$product of $k$ Ginibre matrices, S-Transform of $XX^T$ is $1/(1+z)^k$. This gives inverse MGF of $z/(1+z)^{k+1}$, use series reversion to find that first 2 moments are $1, k,\ldots$. Works for both symmetric uniform and Gaussian entries because they both follow circular law, hence same S-transform | |
| Jul 19, 2023 at 15:59 | comment | added | Yaroslav Bulatov | @IosifPinelis I'm checking see if this follows from techniques in this tutorial | |
| Jul 18, 2023 at 19:18 | history | edited | Yaroslav Bulatov | CC BY-SA 4.0 |
edited tags
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| Jul 18, 2023 at 19:05 | comment | added | Yaroslav Bulatov | @IosifPinelis yes, $A$ and $B$ are independent. I also tried samping $A$, $B$ from the same distribution and it still holds. Purity of product of $A_1\ldots A_k$ is $k+1$ whether we use same distribution (graph) for $A_i$ or different. This works for Uniform(-1,1), Normal, breaks down for Cauchy. | |
| Jul 18, 2023 at 19:02 | comment | added | Iosif Pinelis | So, are $A$ and $B$ independent? Also, did you try sampling for both $A$ and $B$ from the standard normal distribution (or for both $A$ and $B$ from a symmetric uniform distribution)? | |
| Jul 18, 2023 at 18:56 | comment | added | Yaroslav Bulatov | @IosifPinelis edited for clarity | |
| Jul 18, 2023 at 18:54 | history | edited | Yaroslav Bulatov | CC BY-SA 4.0 |
added 133 characters in body
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| Jul 18, 2023 at 18:49 | comment | added | Iosif Pinelis | What is $B$ here? | |
| Jul 18, 2023 at 18:25 | history | asked | Yaroslav Bulatov | CC BY-SA 4.0 |