Suppose I have a real $n\times n$ matrix $\mathbf{C}$ that is Hermitian, positive-definite, and circulant. We know that its eigenvalues $\{\lambda_0,\ldots,\lambda_{n-1}\}$ are extraordinarily nice in that they are positive reals and are the output of the discrete Fourier transform of the top row of $\mathbf{C}$.
Consider the sum $\mathbf{C}+\mathbf{D}$ where $\mathbf{D}=\operatorname{diag}(d_0,\ldots,d_{n-1})$ such that $d_i>0$ for all $i$.
Is there a characterization of eigenvalues of $\mathbf{C}+\mathbf{D}$ in terms of $\{\lambda_0,\ldots,\lambda_{n-1}\}$ and $\{d_0,\ldots,d_{n-1}\}$?
From reading the previous MO questions on the similar topic here and here I understand that the chances of finding a nice characterization are pretty slim. However, I hold tepid hope due to the special structure of my particular problem.
The reason for this inquiry is that eventually I would like to minimize the trace of the inverse $\operatorname{Tr}[(\mathbf{C}+\mathbf{D})^{-1}]$ subject to various constraints on $\{d_0,\ldots,d_{n-1}\}$…
Best Answer
Not sure what you'd accept as a characterization.
Easy enough to work out the case $n=2$. We have $$C=\pmatrix{a&b\cr b&a\cr},\qquad D=\pmatrix{r&0\cr0&s\cr}$$ with $a\gt|b|$ and $r,s\gt0$. The eigenvalues of $C$ are $a\pm b$, the eigenvalues of $C+D$ are $$a+{r+s\over2}\pm\sqrt{\left({r-s\over2}\right)^2+b^2}$$ If we call the eigenvalues of $C$, $\lambda=a+b$ and $\mu=a-b$, the formula above becomes $${\lambda+\mu\over2}+{r+s\over2}\pm\sqrt{\left({r-s\over2}\right)^2+\left({\lambda-\mu\over2}\right)^2}$$ which has some pleasing symmetries.