On the Remarkable Formula for Spectral Distance of Block Southeast Submatrix

Document Type : Research Paper


Arak university of Iran



‎‎‎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 $A \in \mathbb{C}^{n\times n}$ 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 $\rm {Rank\{G_D\}} < n+m-1$‎
‎and given eigenvalues of matrix $\mathcal{M} = D‎ - ‎C A^{-1} B $ as $z_1‎, ‎z_2‎, ‎\cdots‎, ‎z_{m}$ where $|z_1|\ge |z_2|\ge \cdots \ge |z_{m-1}|\ge |z_m|$‎.
Finally, an explicit formula is proven for spectral distance $G_D$ and $G_D_0$ which is expressed by the two last eigenvalues of $\mathcal{M}$.


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