Wavelet and Linear Algebra

Abstract This paper explores the Lyapunov, quadratic matrix, and matrix square root equations under normality assumptions on the system matrix, utilizing the Moore-Penrose inverse ($A^\dagger$) for solvability in singular cases. We provide a detailed historical overview and extend our spectral decomposition method to achieve unique solutions for these equations, building on prior work for invertible matrices. The proposed approach handles singular normal matrices via orthogonality conditions on the kernel, and numerical examples confirm its accuracy and efficiency compared to conventional methods.

Abstract In this paper, for a given set of real numbers such as $\sigma$ with only one positive number and zero summation, we find a distance matrix in which the given set $\sigma$ is its spectrum.
Finally,  we solve special cases of the inverse eigenvalue problem in which the matrix solution is a regular spherical distance matrix.

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 $\mathcal{M}$.