Can one estimate the distribution of eigenvalues of a matrix by its Cauchy/Stieltje transform?

Question

Given a real symmetric $n$ dimensional matrix $A$, with eigenvalues $\lambda_i$ I am defining its Cauchy transform as the function, $f_A(z) = \sum_i \frac{1}{z-\lambda_i}\,$

• Is there any information about the $\lambda_i (A) $s that can be inferred from $f_A(z)$?

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For example I would want an analysis of the following kind :

Say one can estimate the value of $z (A,\epsilon) \gt \max \{ \lambda_i \}$ such that for all $z \geq z (A,\epsilon)$, $f_A(z) \leq \epsilon$. Now suppose one is given a vector $v$ of the same dimension as $A$.

• Is there any connection between $\max \{\lambda(A + vv^T) \} - \max\{ \lambda(A) \}$ and $z(A + vv^T,\epsilon) - z(A,\epsilon)$? (at least for any restricted class of $A$ and $v$?)

Here I picked on $z(A,\epsilon)$ as an example of a property of the Cauchy transform which I feel might be sensitive to how the maximum eigenvalue of $A$ changes under rank-$1$ updates to $A$. But may be this $z(A,\epsilon)$ is not the right quantity - but there is may be some other property of the Cauchy transform which knows about, $\max \{\lambda(A + vv^T) \} - \max\{ \lambda(A) \}$ ?

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If necessary assume that $v = (e_i \pm e_j)$ for some $j > n/2$ and $i<= n/2$.

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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Some possibly related things I saw while searching,

• http://www.people.fas.harvard.edu/~adlam/thesis.pdf
• Intuitive understanding of the Stieltjes transform

Answer 1

Since $A$ is $n\times n$ a hermitian matrix, and thus all the eigenvalues are real, it might be of use to think of $A$ (for $n$ sufficiently large) inducing a probability measure $\mu_A(d\lambda)$. In this sense (and with some more technical poking) we can approximate the Cauchy transform in the original post with the function $$F_A(z) = \int_\mathbb{R} \frac{1}{z - \lambda}\mu_A(d\lambda)$$ This is simply the Stieltjes transform of the measure $\mu_A(d\lambda)$, and you can use the Stieltjes-Perron inversion formula to get $\mu_A$ back.

This is very useful since $F_A$ is analytic on the upper half-plane and goes to zero as $|z| \rightarrow \infty$. We also have the bound $|F_A| \le \frac{\Im{F_A}}{\Im{z}}$. There are a bunch of other results floating around on functions like this, but i have yet to find a comprehensive list.

Answer 2

Your family $(\lambda_i)_{i\in I}$ must reflect all the eigenvalues with their multiplicities, this means, for each $\lambda$ in the spectrum of $A$, the number of times it is repeated in the family $$ n(\lambda)=card(\{i\in I|\lambda_i=\lambda\}) $$ must be its multiplicity (i.e. the multiplicity of $\lambda$ as root of the polynomial $det(\lambda I-A)$)

Then $$ f_A(z)=tr[(zI-A)^{-1}] $$ (this embraces the case when $k\not=\mathbb{C}$)

and you can recover the spectrum (and its multiplicities) from $f_A$ (which is rational) by decomposing it in partial fractions, precisely $$ parfrac(f_A,z,\lambda)=\frac{n(\lambda)}{z-\lambda} $$ Hope it helps.