eigenvalue of symmetric matrix where some diagonal elements dominate

Question

A symmetric matrix $M$ has the following properties:

$$ M_{ii}\gg M_{ij} ~~~~~~ i\neq j ~~~~~~~~~~~~\text{for}~ i>i_0~~~~~~ $$

and all the dominated diagonal elements are equal.

My question is:

Can some of the eigenvalues of $M$ be approximated by making all $M_{ij}=0$ for $i>i_0>j$ and $i<i_0<j$