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I am trying to find out the distribution of a matrix exponential which is a function of a random variable. My mathematics background is very limited and I hope I can receive some help from here.

What I have is a real symmetric matrix $M\in R^{n\times n}$ with $m_{pq}=m_{qp}=\kappa c_{pq}={\kappa}c_{qp}$ when $p\neq q$ and $m_{pq}=-k_p$ when $p=q$, where $k_p$ and $c_{pq}$ are real constants, and $\kappa$ is a Gaussian random variable: $\kappa\sim N(0,\sigma)$ with $|\sigma c_{pq}|\ll|k_p|$. The values of $k_p$ are very close to each other, and $M$ is sparse i.e. $m_{pq}, p\ne q$ are nearly zero everywhere. I need to find out the distribution of the components of the matrix exponential $D=e^{iMt}$, where $i=\sqrt{-1}$, and $t$ is a large positive integer.

Because $|\sigma c_{pq}|\ll|k_p|$, my first attempt is to treat the off-diagonal elements of $M$ as small perturbations and use the first order perturbation theory to compute the eigenvalues and eigenvectors as simple linear functions of $\kappa$. Because the matrix exponential can be obtained by some simple manipulation of the eigenvalues and eigenvectors, I thought I could get the distribution of $D$ easily. However, some of the eigenvectors are found to be very sensitive to the perturbations, probably because $M$ is sparse and with nearly equal diagonal elements. Large error is found from this method.

My next attempt is to separate $M$ into two parts: $M=R+S$, where $R$ is a diagonal matrix with $r_{pq}= m_{pq}$ if $p=q$ and $r_{pq}=0$ if $p\neq q$, and $S$ is the matrix that contains the off-diagonal components of $M$. Because $|\kappa c_{pq}|$ is generally small, $D=e^{iMt}=e^{i(R+S)t}\approx (e^{iR}e^{iS})^t$. The distribution of $U=e^{iR}e^{iS}$ is easy to evaluate. Because: $u_{pq}\approx e^{-ik_{p}}$ if $p=q$ and $u_{pq}\approx i\kappa c_{pq}e^{-ik_p}$ if $p\neq q$, the diagonal components of $U$ are constants,and the off-diagonal components of $U$ follows the imaginary Gaussian distribution. However, although it looks simple, I don't know how to evaluate the distribution of $U^t$.

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