Eigenvalues of a sum of a non-negative symmetric Toeplitz and a non-negative diagonal matrix

Question

I have an $n\times n$ Toeplitz matrix $\mathbf{A}$ that is non-negative and symmetric (that is, $A_{i,j}=A_{j,i}=a_i\geq 0$) and a diagonal matrix $\mathbf{B}=\operatorname{diag}(b_1,b_2,\ldots,b_n)$ where $b_i\geq 0$.

Are there are any theorems/lemmas on the eigenvalues of the sum $\mathbf{A}+\mathbf{B}$? Specifically, I am looking for the upper and lower bounds (or exact results if they exist) on the maximum and minimum eigenvalues, respectively.

Answer 1

You might try to look up the Courant-Fischer (sometimes called Courant Mini-max) Theorem. It does a decent job in certain contexts of estimating maximum eigenvalues.