关于交换扩张的唯一性与计算
On the uniqueness and computation of commuting extensions
AI总结：
本文研究了矩阵交换扩张的唯一性与计算，提出了关于三个及以上矩阵的唯一性定理、适用于 r=4n/3 的高效计算算法及证明其广泛适用性的通用性定理。
摘要：
一个大小为 r 的矩阵元组 (Z_1,...,Z_p) 被称为大小为 n < r 的矩阵元组 (A_1,...,A_p) 的交换扩张，如果 Z_i 两两交换，且每个 A_i 位于 Z_i 的块分解的左上角。这一概念在多个背景下被发现和重新发现，包括代数复杂性理论（在 Strassen 关于张量秩的工作中）、用于构造求积公式的数值分析，以及量子力学中用于研究计算方法和所谓的“量子 Zeno 动力学”。交换扩张也引起了线性代数界的关注。在本文中，我们提出了 3 类结果：(i) 关于三个或更多矩阵的交换扩张唯一性定理。(ii) 计算最小尺寸交换扩张的算法。这些算法在与我们的唯一性定理相同的假设下工作。它们适用于 r=4n/3 的情况，并且显然是此问题中首个适用于 r=n+1 之外的可证明高效的算法。(iii) 一个通用性定理，表明我们的算法和唯一性定理可应用于广泛的输入矩阵。
英文摘要：
A tuple (Z_1,...,Z_p) of matrices of size r is said to be a commuting extension of a tuple (A_1,...,A_p) of matrices of size n <r if the Z_i pairwise commute and each A_i sits in the upper left corner of a block decomposition of Z_i. This notion was discovered and rediscovered in several contexts including algebraic complexity theory (in Strassen's work on tensor rank), in numerical analysis for the construction of cubature formulas and in quantum mechanics for the study of computational methods and the study of the so-called "quantum Zeno dynamics." Commuting extensions have also attracted the attention of the linear algebra community. In this paper we present 3 types of results: (i) Theorems on the uniqueness of commuting extensions for three matrices or more. (ii) Algorithms for the computation of commuting extensions of minimal size. These algorithms work under the same assumptions as our uniqueness theorems. They are applicable up to r=4n/3, and are apparently the first provably efficient algorithms for this problem applicable beyond r=n+1. (iii) A genericity theorem showing that our algorithms and uniqueness theorems can be applied to a wide range of input matrices.
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