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Suppose $M$ is an $n\times n$ matrix with IID random entries drawn from $\mathcal{D}$ and $\sigma$ is the vector of its singular values. Define purity of $M$ as

$$\rho(M)=\frac{n \sum_i \sigma_i^4}{\left(\sum_i \sigma_i^2\right)^2}$$

Take square matrices $A$, $B$ with entries sampled IID from distributions $\mathcal{D}_1$, $\mathcal{D}_2$ respectively. The following appears to hold empirically for standard normal $\mathcal{D}_1$ and symmetric uniform $\mathcal{D}_2$ and $n=1000$

$$\rho(AB)\approx \rho(A)+\rho(B)-1$$

Is this a well-known result? Any pointers to the literature appreciated!

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  • $\begingroup$ What is $B$ here? $\endgroup$
    – Iosif Pinelis
    Commented Jul 18, 2023 at 18:49
  • $\begingroup$ @IosifPinelis edited for clarity $\endgroup$
    – Yaroslav Bulatov
    Commented Jul 18, 2023 at 18:56
  • $\begingroup$ 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)? $\endgroup$
    – Iosif Pinelis
    Commented Jul 18, 2023 at 19:02
  • $\begingroup$ @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. $\endgroup$
    – Yaroslav Bulatov
    Commented Jul 18, 2023 at 19:05
  • $\begingroup$ @IosifPinelis I'm checking see if this follows from techniques in this tutorial $\endgroup$
    – Yaroslav Bulatov
    Commented Jul 19, 2023 at 15:59

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