A Quiver Invariant Theoretic Approach to Radial Isotropy and the Paulsen Problem for Matrix Frames

IF 1.6 2区 数学 Q2 MATHEMATICS, APPLIED
C. Chindris, Jasim Ismaeel
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引用次数: 0

Abstract

In this dissertation, we view matrix frames as representations of quivers and study them within the general framework of Quiver Invariant Theory. We are particularly interested in radial isotropic and Parseval matrix frames. Using methods from Quiver Invariant Theory [CD21], we first prove a far-reaching generalization of Barthe's Theorem [Bar98] on vectors in radial isotropic position to the case of matrix frames (see Theorems 5.13(3) and 4.12). With this tool at our disposal, we generalize the Paulsen problem from frames (of vectors) to frames of matrices of arbitrary rank and size extending Hamilton-Moitra's upper bound [HM18]. Specifically, we show in Theorem 5.20 that for any given ε-nearly equal-norm Parseval frame F of n matrices with d rows there exists an equal-norm Parseval frame W of n matrices with d rows such that dist^2 (F,W) [less than or equal to] 46[epsilon]d^2. Finally, in Theorem 5.28 we address the constructive aspects of transforming a matrix frame into radial isotropic position which extend those in [Bar98, AKS20].
矩阵框架径向各向同性和Paulsen问题的颤振不变量理论研究
本文将矩阵框架视为颤振的表示,并在颤振不变性理论的一般框架内对其进行了研究。我们对径向各向同性和Parseval矩阵框架特别感兴趣。利用Quiver Invariant Theory [CD21]中的方法,我们首先证明了Barthe定理[Bar98]在径向各向同性位置上对矩阵框架的推广(见定理5.13(3)和4.12)。利用这个工具,我们将Paulsen问题从(向量的)框架推广到扩展Hamilton-Moitra上界的任意秩和大小的矩阵框架[HM18]。具体地,我们在定理5.20中证明了对于任意给定的ε-近似等范数Parseval坐标系F (n个d行矩阵)存在一个包含n个d行矩阵的等范数Parseval坐标系W,使得dist^2 (F,W)[小于或等于]46[ε]d^2。最后,在定理5.28中,我们讨论了将矩阵框架转换为径向各向同性位置的建设性方面,扩展了[Bar98, AKS20]中的内容。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
2.20
自引率
0.00%
发文量
19
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