Perron相似性与非负逆特征值问题
Perron similarities and the nonnegative inverse eigenvalue problem
AI总结：
本文发展复Perron相似性理论，通过研究I型Karpelevich弧实现矩阵的Perron相似性生成大量可实现谱，验证4×4随机矩阵谱的几何特性，还得到DFT矩阵等相关多胞体的极值性质，推进非负逆特征值问题研究。
摘要：
长期存在的非负逆特征值问题（nonnegative inverse eigenvalue problem，NIEP）旨在确定哪些复数多重集可作为元素非负矩阵的谱。尽管已有一些著名的必要条件，但NIEP的完整解决方案仍远未明晰。若一个可逆矩阵可将某个不可约非负矩阵对角化，则称其为Perron相似矩阵。Johnson与Paparella已建立了实Perron相似性的理论，本文则全面发展了复Perron相似性的理论。每个Perron相似性都对应一个非平凡的多面锥与可实现谱多胞体（可视为复欧几里得空间中的向量）。这些凸集的极值点数量有限，确定每个Perron相似性对应的极值点即可解决可对角化NIEP——这是整个问题的核心组成部分。通过研究I型Karpelevich弧的某些实现矩阵的Perron相似性，可生成给定正整数下的大量可实现谱。本文通过生成4×4随机矩阵谱的近乎完整几何表示验证了这一点。与Karpelevich区域类似，研究表明由随机矩阵谱构成的复欧几里得空间子集是紧的且为星形的。本文定义了该集合的极值元素，并证明其位于边界上。研究还表明，离散傅里叶变换（discrete Fourier transform，DFT）矩阵对应的多面锥与凸多胞体分别对应其行的锥包与凸包。本文还对DFT矩阵的多重Kronecker积、以及DFT矩阵与Walsh矩阵的多重Kronecker积建立了类似结果。这些多胞体对于NIEP具有重要意义，因为它们在由随机矩阵谱构成的区域中是极值的。
英文摘要：
The longstanding nonnegative inverse eigenvalue problem (NIEP) is to determine which multisets of complex numbers occur as the spectrum of an entry-wise nonnegative matrix. Although there are some well-known necessary conditions, a solution to the NIEP is far from known. An invertible matrix is called a Perron similarity if it diagonalizes an irreducible, nonnegative matrix. Johnson and Paparella developed the theory of real Perron similarities. Here, we fully develop the theory of complex Perron similarities. Each Perron similarity gives a nontrivial polyhedral cone and polytope of realizable spectra (thought of as vectors in complex Euclidean space). The extremals of these convex sets are finite in number, and their determination for each Perron similarity would solve the diagonalizable NIEP, a major portion of the entire problem. By considering Perron similarities of certain realizing matrices of Type I Karpelevich arcs, large portions of realizable spectra are generated for a given positive integer. This is demonstrated by producing a nearly complete geometrical representation of the spectra of $4 \times 4$ stochastic matrices. Similar to the Karpelevich region, it is shown that the subset of complex Euclidean space comprising the spectra of stochastic matrices is compact and star-shaped. Extremal elements of the set are defined and shown to be on the boundary. It is shown that the polyhedral cone and convex polytope of the discrete Fourier transform (DFT) matrix corresponds to the conical hull and convex hull of its rows, respectively. Similar results are established for multifold Kronecker products of DFT matrices and multifold Kronecker products of DFT matrices and Walsh matrices. These polytopes are of great significance with respect to the NIEP because they are extremal in the region comprising the spectra of stochastic matrices.
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