# Properties of eigenvalues and eigenvectors of a particular random matrix

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 $$\mathbf{Q}$$?