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.

翻译：本文研究矩阵交换扩张的唯一性与计算问题。设 (Z_1,...,Z_p) 是尺寸为 r 的矩阵元组，若其满足两两交换条件，且每个 A_i 位于 Z_i 分块分解的左上角，则称该元组为尺寸 n < r 的矩阵元组 (A_1,...,A_p) 的交换扩张。这一概念在代数复杂度理论（Strassen 关于张量秩的研究）、数值分析（构造求积公式）以及量子力学（计算方法研究与所谓"量子芝诺动力学"分析）等多个领域被独立发现。交换扩张问题亦引起了线性代数界的关注。本文提出三类结果：(i) 针对三个及以上矩阵的交换扩张唯一性定理；(ii) 最小尺寸交换扩张的计算算法。该算法在唯一性定理相同假设下运行，可适用于 r=4n/3 以下情形，据我们所知，这是首个在此假设范围（超越 r=n+1）内具有可证效率的算法；(iii) 一般性定理表明，本文算法与唯一性定理可广泛应用于各类输入矩阵。