I am working with two random matrices, $\mathbf{Z}$ and $\mathbf{H}$:

1. $\mathbf{Z}$ is an $N \times K$ matrix with entries sampled i.i.d. from a Bernoulli distribution: $Z_{nk} \sim \mathrm{Bernoulli}(\nu)$.
2. $\mathbf{H}$ is a $K \times K$ matrix with entries sampled i.i.d. from a Gaussian distribution: $\mathbf{H} \sim \mathcal{N}(\boldsymbol{\mu}_H, \boldsymbol{\Sigma}_H)$.

I've defined a matrix product as:
$ \kappa = \mathbf{Z}\mathbf{H}\mathbf{Z}^T $

Given that $\mathbf{Z}$ and $\mathbf{H}$ are independent, so their expectation is equal to zzero, I'm interested in computing:$\mathbb{E}[\kappa^2]$ w.r.t $\mathbf{Z}$ and $\mathbf{H}$ variables.

Any suggestions on how to approach this problem would be greatly appreciated. Thank you!