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

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 $[a_i,b_i]$?

What is the distribution of $$\frac{||\sum_i x_i M_i||_1}{||\sum_i x_i M_i||_2}$$