对角受迫系统与矩阵环的谱特征
Diagonally forced systems and the spectral signature of matrix cycles
AI总结：
本文研究对角元素为负的矩阵经变量 $σ$ 缩放后的稳定性临界值，建立特征多项式根与缩放特征值的等价关系，并据此分离主导反馈环、提出矩阵的内在谱特征。
摘要：
给定任意对角元素为负的方阵 $\mathbf{M}$，并将其乘以变量 $σ$，我们希望找到使 $\mathbf{M}_σ$ 的特征值恰好为零的最小 $σ$。由 Gershgorin（盖尔戈林）定理可知，通过使 $σ$ 足够大，可以让 $\mathbf{M}_σ$ 稳定。我们证明了一种关系，它从解析上确定何时以及如何能够找到使最大特征值恰好为零的 $σ$ 值。在此过程中，我们证明了 $\mathbf{M}_σ$ 的特征多项式的根与对 $\mathbf{M}$ 进行尺度缩放操作所产生的特征值等价。此外，通过特征多项式，我们能够分离出构成矩阵元素的主导反馈环；在 $σ$ 的作用下，这些反馈环确保系统的稳定性。随后，我们利用特征多项式系数中的稳定化与失稳化环，探索了与任意矩阵相关的一种内在谱特征，该特征基于矩阵各元素的大小和符号（或零）。
英文摘要：
Given any square matrix, $\mathbf{M}$, whose diagonal elements are negative, and which are multiplied by a variable, $σ$, we wish to find the minimal $σ$ such that the eigenvalue of $\mathbf{M}_σ$ is exactly zero. By Gershgorin, we know that $\mathbf{M}_σ$ can be made stable by making $σ$ large enough. We prove a relation which analytically determines when and how we are able to find the value of $σ$ such that the maximal eigenvalue is exactly zero. In so doing, we prove the equivalence of the roots of the characteristic polynomial of $\mathbf{M}_σ$ and the eigenvalues that arise from a scaling operation on $\mathbf{M}$. Further, through the characteristic polynomial, we are able to isolate the dominant feedback cycles comprising the elements of the matrix which, under the action of $σ$, ensures the stability of the system. We then explore, using the stabilising and destabilising cycles within the coefficients of the characteristic polynomial, an intrinsic spectral signature associated with any square matrix based on the sign (or zero) of its respective elements.
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