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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 $\mathbf{Q}$?

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This is addressed in this nice paper:

Hachem, Walid; Loubaton, Philippe; Najim, Jamal, Deterministic equivalents for certain functionals of large random matrices, Ann. Appl. Probab. 17, No. 3, 875-930 (2007). ZBL1181.15043.

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