Perron-Frobenius定理的一个新证明：变分方法
A new proof of the Perron-Frobeniuos theorem, a variational approach
AI总结：
本文用变分方法给出Perron-Frobenius定理的新证明，引入拟特征值概念，推广Weyl不等式并证明Perron根在任意扰动下的稳定性。
摘要：
我们采用变分方法发展Perron-Frobenius理论，并将其推广到任意矩阵集合，包括既不可约也非本质正定的矩阵，以及非保持锥。我们引入了一个称为矩阵拟特征值的新概念，它在变量的正交变换下不变，并具有多种有用性质，例如确定矩阵特征值实部的最大值。我们将Weyl不等式推广到任意矩阵集合，并证明了不可约非负矩阵的Perron根在任意扰动下的新稳定性结果。此外，我们还获得了特征值集合及其实部范围的新类型估计。
英文摘要：
We develop the Perron-Frobenius theory using a variational approach and extend it to a set of arbitrary matrices, including those that are neither irreducible nor essentially positive, and non-preserved cones. We introduce a new concept called a quasi-eigenvalue of a matrix, which is invariant under orthogonal transformations of variables, and has various useful properties, such as determining the largest value of the real parts of the eigenvalues of a matrix. We extend Weyl's inequality for the eigenvalues to the set of arbitrary matrices and prove the new stability result to the Perron root of irreducible nonnegative matrices under arbitrary perturbations. As well as this, we obtain new types of estimates for the ranges of the sets of eigenvalues and their real parts.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
