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_۰} = \begin{pmatrix}‎ ‎A & B \\‎ ‎C & D_۰‎ ‎\end{pmatrix} to set of block normal matrix G_{D} (as same as G_{D_۰} except block D which is replaced by block D_۰)‎, ‎in which A \in \mathbb{C}^{n\times n} is invertible‎, ‎ B \in \mathbb{C}^{n\times m}‎, ‎C \in \mathbb{C}^{m\times n} and D \in \mathbb{C}^{m\times m} with \rm {Rank\{G_D\}} < n+m-۱‎ ‎and given eigenvalues of matrix \mathcal{M} = D‎ - ‎C A^{-۱} B as z_۱‎, ‎z_۲‎, ‎\cdots‎, ‎z_{m} where |z_۱|\ge |z_۲|\ge \cdots \ge |z_{m-۱}|\ge |z_m|‎. Finally, an explicit formula is proven for spectral distance G_D and G_D_۰ which is expressed by the two last eigenvalues of \mathcal{M}.

Keywords:

Eigenvalues , Normal matrix , Distance norm