Principal minor resolvent trace bound ($|N^{-1}\mathrm{Tr}{G}_X-N^{-1}\mathrm{Tr}{G}_{X^{(i)}}|\le c/(N|\mathrm{Im}z|)$)

The eigenvalue interlacing of a Hermitian matrix and a principal minor bounds the difference of their resolvent traces by a constant times $|\mathrm{Im}z{|}^{-1}$. When both traces are normalized by the original dimension $N$, the bound acquires $N^{-1}$. One proof writes the difference using the interlacing counting functions, whose difference is at most one, and bounds the integral of $|t-z{|}^{-2}$ by $\pi /|\mathrm{Im}z|$. This gives the admissible absolute constant $c=\pi$.