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Suppose I have a tridiagonal square matrix $A$ of some nice form, for which I know the eigenvalues $\lambda_1<\dots<\lambda_n$. $A$ is also essentially nonnegative (nonnegative everywhere except main diagonal).

What can I say about the sum of $A$ and a diagonal matrix $B=\text{diag}(x_1,\dots,x_n)$, for which I know only about sum of it's elements?

Some equalities, maybe inequalities with eigenvalues of the first matrix and values of the second? Special interest is in the maximal eigenvalue

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I'm not sure I understand the question completely, but the SUM of the diagonal entries certainly can't give anything (in general) on individual eigenvalues ($A=0$). –  Christian Remling Apr 27 at 23:38
    
assuming a symmetric tridiagonal, isn't the majorization relation $\lambda(A+B)\prec \lambda(A)+\lambda(B)$ sufficient? –  Suvrit Apr 28 at 0:34
    
@ChristianRemling that is all I know: $A$ is tridiagonal, $B=\text{diag}(x_1,\dots,x_n),\ \sum_{1}^{n} x_i = R$. But any interesting inequality which contain x_i and eigenvalues of $A$ would be a great step for me too –  Daniel Apr 29 at 1:09
    
@Suvrit I wonder if there is some more interesting property based on tridiagonality and nonnegativity of $A$ and diagonality of $B$ –  Daniel Apr 29 at 1:17

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