on the remarkable formula for spectral distance of block southeast submatrix

‎‎‎this paper presents a remarkable formula for spectral distance of a given block normal matrix $g_{d_0} = begin{pmatrix}‎ ‎a & b ‎ ‎c & d_0‎ ‎end{pmatrix} $ to set of block normal matrix $g_{d}$ (as same as $g_{d_0}$ except block $d$ which is replaced by block $d_0$)‎, ‎in which $a in mathbb{c}^{ntimes n}$ is invertible‎, ‎$ b in mathbb{c}^{ntimes m}‎, ‎c in mathbb{c}^{mtimes n}$ and $d in mathbb{c}^{mtimes m}$ with $rm {rank{g_d}} < n+m1$‎ ‎and given eigenvalues of matrix $mathcal{m} = d‎ ‎c a^{1} b $ as $z_1‎, ‎z_2‎, ‎cdots‎, ‎z_{m}$ where $|z_1|ge |z_2|ge cdots ge |z_{m1}|ge |z_m|$‎. finally, an explicit formula is proven for spectral distance $g_d$ and $g_d_0$ which is expressed by the two last eigenvalues of $mathcal{m}$.