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Say $A$ is a symmetric matrix of $n$ dimensions. Then let the ``resolvent" of $A$ be the matrix valued function $R_A(z) = \frac{1}{z-A}$ and its Cauchy transform be the real valued function $C_A(z) = Tr[R_A(z) ]$.

  • I want to compare between the numbers $[R_A(z)]_{ii}$ and $\frac{1}{n} C_A(z)$.

    Do we know as to when is which of them larger and by how much?

    Like given a $A$ and a $i$ I want to understand when it will be true that, $[R_A(z)]_{ii} \leq \frac{1}{n} C_A(z)$ ?

If necessary assume that $z > \lambda_{max}(A)$

If necessary assume that $A$ is constructed as follows : First take the matrix $D - Ad$ where $D$ is the diagonal matrix of degrees of some bi-partite graph and $Ad$ is its adjacency matrix. Then flip some of the off-diagonal $-1$ entries of $D-A$ to $1$ keeping the entire thing symmetric.

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  • $\begingroup$ Robert Israel's answer is the best possible: There is always some index $i$ which satisfies the inequality for trivial reasons. And no more can be said because the whole problem is invariant under base change, so we can reindex the rows and columns (i.e. conjugate $A$ by a permutation matrix) such that the index where the inequality holds is 1 or 42 or 4711 or any other index you want. $\endgroup$
    – Johannes Hahn
    Commented Jan 26, 2018 at 13:00
  • $\begingroup$ Since this question keeps getting bumped, would you consider accepting Robert Israel's answer? $\endgroup$
    – Yemon Choi
    Commented Feb 25, 2018 at 15:34

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The trace is the sum of the diagonal elements, so (when $z$ is real) it's always true for at least one $i$, and the only way it can be true for all of them is that all diagonal elements are equal. For example, this is the case for the matrix $$ A = \pmatrix{3 & 0 & 1 & -1\cr 0 & 3 & 1 & -1\cr 1 & 1 & 3 & 0\cr -1 & -1 & 0 & 3\cr} $$

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  • $\begingroup$ Seems like I phrased the question badly! Sorry. I edited it. Say I fix an $i$ then I want to know as to when can the inequality be true. $\endgroup$
    – Student
    Commented Jun 3, 2015 at 2:56
  • $\begingroup$ I want to say something like this - that this inequality will always be true some $i$ for which the diagonal entry of $A$ is minimum. Something like this... $\endgroup$
    – Student
    Commented Jun 3, 2015 at 2:58
  • $\begingroup$ The A in your example doesn't satisfy the 3rd criteria of mine when I list the wanted properties of $A$. $\endgroup$
    – Student
    Commented Jun 3, 2015 at 2:59
  • $\begingroup$ You can add a constant multiple of the identity to make it satisfy that. OK, I'll edit. $\endgroup$
    – Robert Israel
    Commented Jun 3, 2015 at 3:46
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    $\begingroup$ So you mean the number of off-diagonal nonzero elements in the row/column. OK, change the $3$'s to $2$'s. $\endgroup$
    – Robert Israel
    Commented Jun 3, 2015 at 15:17

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