Let $\mathbf{A}$ be a random matrix of size $N\times N$ and let $\mathbf{D}$ be any diagonal matrix.
If we denote $\{\alpha_i\}$ the eigenvalues of $\mathbf{A}$ ($\alpha_i \in \mathbb{C}$) and $\{\lambda_i\}$ the eigenvalues of $\mathbf{D}$: Is there an explicit formula/equation to represent the eigenvalues of $\mathbf{M}=\mathbf{A}+\mathbf{D}$?
When $\mathbf{D}=x\cdot\mathbf{I}$ the identity matrix, the eigenvalues will simply be shifted by a factor of $x$, but when the elements of $\mathbf{D}$ are not identical I am not quite sure what to expect?
Any ideas or advice is appreciated!
Well in general the only time when the eigenvalues of a sum of matrix can be linked to the eigenvalues of the summed matrices is when they share common eigenvectors. Otherwise, there really isn't much to say...
Here it means that you'll be able to find $p$ eigenvalues of $M$ if $p$ of the columns of $A$ have only one non-zero coefficient. Indeed, the eigenvectors of $D$ are $e_1, e_2,...,e_n$ where $e_i$ are vectors with their i-th coefficient equal to one and others equal to zero. Outside of this particular case I don't think there exists a simple link between the eigenvalues of $M$, $A$,and $D$.