Frobeinus norm of multiplication of two complex Gaussian distributed matrices

Question

There are two complex Gaussian distributed matrices, $\mathbf{A}\in \mathbb{C}^{L\times M}$ and $\mathbf{B}\in \mathbb{C}^{N\times M}$.

The elements of $\mathbf{A}$ and $\mathbf{B}$ are followed i.i.d. complex Gaussian distribution with mean 0 and variances $\sigma_A^2$ and $\sigma_B^2$, respectively, i.e., $a_{ij}\sim \mathcal{CN}(0,\sigma_A^2)$ and $b_{ij}\sim \mathcal{CN}(0,\sigma_B^2)$ for $\forall i,j.$

I want to find the expectation of $||\mathbf{A}\mathbf{B}^H||_F^2$.

So, how can I express the closed form of $\mathbb{E}\big[||\mathbf{A}\mathbf{B}^H||_F^2\big]$?

Answer 1

What is $\textbf{I}$? a matrix with all coefficients equal to $1$? What is $B^H$? Hermitian conjugate? So we are with complex Gaussian variables? What is $|AB^H|? $| A determinant? So we are in the case $L=N$?

Answer 2

Therefore you ask for $$E(\mathrm{trace}(AB^HBA^H))=E(\mathrm{trace}(A C A^T))$$ where $C=E(B^HB).$ Without loss of generality you may assume $\sigma_A=\sigma_B=1.$ You will multiply by $\sigma^2_A\times \sigma_B^2$ the final result.

Actually $C$ is proportional to the identity matrix as well as $E((A C A^T)).$ Since I am not sure of your definition of a standard complex Gaussian variable, I leave to you the remaining simple calculation!