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Given basis $M_1,M_2\dotsc,M_{d^2}$ in $\mathbb C^{d\times d}$, we consider $$\sum_i x_i M_i$$ for random variables $x_i$.

What is the distribution of $$\lVert\sum_i x_i M_i\rVert_1=\sum \sigma_k?$$ Here $\sigma_k$s are singular values of $\sum_i x_i M_i$.

What is the distribution of $$\frac{\lVert\sum_i x_i M_i\rVert_1}{\lVert\sum_i x_i M_i\rVert_2}?$$

What is the eigenvalue distribution (the vector of the eigenvalues) of $\sum_i x_i M_i$, if $x_i$ are independent Gaussian $\mathcal{N}(\mu_i,\sigma_i^2)$? If $x_i$ are uniform distribution in $[a_i,b_i]$?

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  • $\begingroup$ Do you mean multiset rather than vector of eigenvalues? $\endgroup$
    – Richard Stanley
    Commented Aug 13, 2022 at 3:05
  • $\begingroup$ I am interested in the vector of eigenvalues. $\endgroup$
    – gondolf
    Commented Aug 13, 2022 at 11:23
  • $\begingroup$ In what order do you put the eigenvalues in the vector? $\endgroup$
    – Richard Stanley
    Commented Aug 13, 2022 at 15:24
  • $\begingroup$ If you want to map a $d\times d$ complex matrix $M$ to a vector $v$ in $\mathbb{C}^d$ that records the eigenvalues of $M$, then we should take $v=(e_1,\dots,e_d)$, where $e_i$ is the $i$th elementary symmetric function of the eigenvalues. $\endgroup$
    – Richard Stanley
    Commented Aug 14, 2022 at 2:43
  • $\begingroup$ @Stanley Yes. Thank you very much for clarifying this point. I think I am more interested in the distribution of the sum of singular values. $\endgroup$
    – gondolf
    Commented Aug 14, 2022 at 23:38

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