基于线性代数的对称矩阵束到奇异点的结构距离估计
Estimation to structured distances to singularity for matrix pencils with symmetry structures: A linear algebra-based approach
摘要：
研究了正则矩阵束$A+sE$到奇异点的结构距离，其中$(A,E)\in \mathbb S \subseteq (\mathbb C^{n,n})^2$ .这包括厄米、斜厄米、$*$ -偶、$*$ -奇、$*$ -回文、T-回文和耗散哈密顿束。我们提出了一个纯粹的基于线性代数的方法来推导出显式的可计算公式的距离最近的结构铅笔$(A-Δ_A)+s(E-Δ_E)$，使$A-Δ_A$和$E-Δ_E$有一个共同的零向量。然后，我们得到一个家庭的可计算的下界的非结构化和结构化的距离奇异性。数值实验表明，在许多情况下，结构化和非结构化的距离之间存在显着差异。这种方法扩展到具有更高次数的结构化矩阵多项式。
英文摘要：
We study the structured distance to singularity for a given regular matrix pencil $A+sE$, where $(A,E)\in \mathbb S \subseteq (\mathbb C^{n,n})^2$. This includes Hermitian, skew-Hermitian, $*$-even, $*$-odd, $*$-palindromic, T-palindromic, and dissipative Hamiltonian pencils. We present a purely linear algebra-based approach to derive explicit computable formulas for the distance to the nearest structured pencil $(A-Δ_A)+s(E-Δ_E)$ such that $A-Δ_A$ and $E-Δ_E$ have a common null vector. We then obtain a family of computable lower bounds for the unstructured and structured distances to singularity. Numerical experiments suggest that in many cases, there is a significant difference between structured and unstructured distances. This approach extends to structured matrix polynomials with higher degrees.
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