Let $\mathbf{A}$ be a given $n \times m$ matrix with positive entries, and $\mathbf{B}_{n\times m}$ be a random i.i.d complex Gaussian matrix with unit variance. Assume that $\mathbf{C}$ is the Hadamard (element-wise) product of these two matrices, i.e.,
$$\mathbf{C} = \mathbf{A} \odot \mathbf{B}.$$
What are the propositions about the eigenvalues and eigenvectors of the matrix $\mathbf{C}^{\mathrm{H}}\mathbf{C}$, where $\mathbf{C}^{\mathrm{H}}$ is the conjugate transpose of $\mathbf{C}$?
For example, for the following diagonalization
$$\mathbf{C}^{\mathrm{H}}\mathbf{C}=\mathbf{Q}^{\mathrm{H}}\mathbf{\Lambda}\mathbf{Q},$$
where $\mathbf{Q}$ is the unitary matrix and $\mathbf{\Lambda}$ is the eigenvalues diaonal matrix, matrices $\mathbf{Q}$ and $\mathbf{\Lambda}$ are independent? What can we say about the distribution of $\mathbf{\Lambda}$? Is there any proposition about the variance of the elements of $\mathbf{Q}$?
