On the uniqueness and computation of commuting extensions

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications Pub Date : 2024-10-09 DOI:10.1016/j.laa.2024.10.004

Pascal Koiran

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引用次数: 0

Abstract

A tuple

(Z_{1}, \dots, Z_{p})

of matrices of size

r \times r

is said to be a commuting extension of a tuple

(A_{1}, \dots, A_{p})

of matrices of size

n \times n

if the

Z_{i}

pairwise commute and each

A_{i}

sits in the upper left corner of a block decomposition of

Z_{i}

(here, r and n are two arbitrary integers with

n < r

). 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 $r = 4 n / 3$ , 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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关于换向扩展的唯一性和计算

大小为 r×r 的矩阵元组 (Z1，...，Zp) 是大小为 n×n 的矩阵元组 (A1，...，Ap) 的换向扩展，如果 Zi 成对换向，并且每个 Ai 都位于 Zi 的块分解的左上角（这里，r 和 n 是两个任意整数，n<r）。这一概念在多个领域被发现和重新发现，包括代数复杂性理论（在斯特拉森关于张量秩的研究中）、用于构造立方公式的数值分析，以及用于研究计算方法和所谓 "量子芝诺动力学 "的量子力学。换元扩展也引起了线性代数界的关注。在本文中，我们提出了三类结果：(i) 三个或更多矩阵的换元扩展唯一性定理；(ii) 计算最小尺寸换元扩展的算法。这些算法的假设条件与我们的唯一性定理相同。(iii)通用性定理表明我们的算法和唯一性定理可以应用于广泛的输入矩阵。

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

Linear Algebra and its Applications 数学-应用数学

CiteScore

2.20

自引率

9.10%

发文量

333

审稿时长

13.8 months

期刊介绍： Linear Algebra and its Applications publishes articles that contribute new information or new insights to matrix theory and finite dimensional linear algebra in their algebraic, arithmetic, combinatorial, geometric, or numerical aspects. It also publishes articles that give significant applications of matrix theory or linear algebra to other branches of mathematics and to other sciences. Articles that provide new information or perspectives on the historical development of matrix theory and linear algebra are also welcome. Expository articles which can serve as an introduction to a subject for workers in related areas and which bring one to the frontiers of research are encouraged. Reviews of books are published occasionally as are conference reports that provide an historical record of major meetings on matrix theory and linear algebra.