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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.

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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.

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    Yeah, this is interesting. Thanks for the reminder about the Courant-Fischer. Btw, $\mathbf{A}+\mathbf{B}$ is not Toeplitz, as the diagonal entries aren't the same, however, it is still symmetric. Toeplitz matrices actually do have nice eigenvalue behavior in the limit, since, as their size increases, they get closer to circulant matrices, whose eigenvalues are just the DFT of the top row. The [reference I cited](http://ee.stanford.edu/~gray/toeplitz.pdf) in my earlier comment covers that.2012-04-04