Suppose we are interested in a matrix **A** $\in \mathcal{C}^{N\times N}$, which has elements

$A_{ij} = a_{ij} b_{ij}$

where the elements $a_{ij}$ are real and symmetric. The elements $b_{ij} = e^{i\phi_{ij}}$ are complex numbers of unit length. Now, while the eigendecomposition of $a_{ij}$ possesses convenient properties (real eigenvalues and orthogonal basis), the eigendecomposition of the full matrix **A**, modified by the matrix $b_{ij}$, is of interest.

Are there any useful statements we can make about the eigenspectrum of **A**? I realize that a closed-form answer is not likely in the general case; however, any restrictions of interest would be good to know as well.