# On the Remarkable Formula for Spectral Distance of Block Southeast Submatrix

Document Type: Research Paper

Authors

Arak university of Iran

10.22072/wala.2018.87428.1174

Abstract

‎‎‎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 $\mathcal{M}$.

Keywords

### References

[1] Kh.D. Ikramov and A.M. Nazari, On a remarkable implication of the Malyshev formula, Dokl. Akad. Nauk.,

385 (2002) 599-600.

[2] J.-M. Gracia and F.E. Velasco, Nearesr southeast submatrix that makes multiple a prescribed eigenvalue.

Part 1, Linear Algebra Appl., 430(4) (2009), 1196-1215.

[3] A. Nazari and  A. Nezami, Computational aspect to the nearest southeast submatrix that makes multiple a

prescribed eigenvalue, J. Linear Topol. Algebra, 6(1) (2017), 67-72.