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Is there any analytic expression for summation of eigen-values of a tri-diagonal matrix which are smaller than a constant value? Or even a rough approximation for it. How about case of a general matrix. Considering a known matrix H with eigen values of $\epsilon_i$ (which we don't want to calculate directly by solving eigen-value problem) How I can find an approximate for following according to matrix elements:

$\Sigma_{(\epsilon_i < C )}(\epsilon_i) = \Sigma_i(\epsilon_i * \Theta(C - \epsilon_i))$

Where $\Theta$ is step function and C is a constant.

When C $\to \infty$ the answer is obvious

$\lim_{C \to \infty} \Sigma_{(\epsilon_i < C > )}(\epsilon_i) = trace(H)$

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    $\begingroup$ What's wrong with just using the trace of the matrix? $\endgroup$
    – Casteels
    Commented Feb 19, 2013 at 22:58
  • $\begingroup$ @Casteels: I made a mistake, not all eigen-values; just eigen values smaller than a constant. $\endgroup$
    – Hesam
    Commented Feb 20, 2013 at 2:11
  • $\begingroup$ Is the matrix also Toeplitz? $\endgroup$
    – Rodrigo de Azevedo
    Commented Sep 7, 2016 at 18:58

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