I have a positive definite matrix $n\times n$ matrix $\mathbf{L}$. This matrix is the combinatorial Laplacian of a graph.
We can compute the expected value of the Laplacian as $\mathbb{E}[\mathbf{L}]$ as in the graph model considered, links in the graph appear independently with probability $p_{ij}$. The expected Laplacian has $\sum_k p_{ik}$ on the diagonal and $-p_{ij}$ off diagonal elements.
Now I need to compute the following quantity:
$$ \mathbb{E}\left[ \frac{e^{-\beta \mathbf{L}}}{\textrm{Tr}[e^{-\beta \mathbf{L}}]}\right] $$ where the exponential is intended as the matrix exponential and $\mathrm{Tr}$ is the trace.
I understand that the Jensen inequality can be correctly applied to the denominator $\textrm{Tr}[e^{-\beta \mathbf{L}}]$ as this is a convex scalar function and one gets $\mathbb{E}\left[\textrm{Tr}[e^{-\beta \mathbf{L}}]\right] \geq \mathrm{Tr}\left[ e^{-\beta \mathbb{E}[\mathbf{L}]}\right]$.
However I cannot find references on how to apply Jensen inequality to matrix valued functions. Is it correct to conclude that:
$$ \mathbb{E}\left[ \frac{e^{-\beta \mathbf{L}}}{\textrm{Tr}[e^{-\beta \mathbf{L}}]}\right] \geq \frac{e^{-\beta E[\mathbf{L}]}}{\textrm{Tr}[e^{-\beta E[\mathbf{L}]}]} $$
Is it possible to show this using the Taylor expansion of the exponential or there
It is not possible to apply Jensen inequality because the matrix exponential is not an operator convex function. To see that, look at the Bhatia book "Matrix analysis".