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On the uniqueness and computation of commuting extensions

Pascal Koiran

TL;DR

This work studies commuting extensions of tuples of matrices, providing conditions for essential uniqueness when extending at least three matrices, and presenting polynomial-time algorithms to compute minimal-size extensions up to $r\le 4n/3$. The core approach recasts commutativity via block decompositions and rank-based subspace identities, enabling constructive recovery of the extension blocks and reduction to linear systems. A central contribution is a genericity theorem showing that the required rank/dimension conditions hold for a broad class of inputs, making the algorithms widely applicable. The results have potential impact across algebraic complexity, numerical cubature, and quantum dynamics by enabling reliable construction and classification of commuting extensions. The paper also discusses existence and non-uniqueness phenomena, and outlines directions for extending the theory to the two-matrix case and beyond $r=4n/3$.

Abstract

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.

On the uniqueness and computation of commuting extensions

TL;DR

This work studies commuting extensions of tuples of matrices, providing conditions for essential uniqueness when extending at least three matrices, and presenting polynomial-time algorithms to compute minimal-size extensions up to . The core approach recasts commutativity via block decompositions and rank-based subspace identities, enabling constructive recovery of the extension blocks and reduction to linear systems. A central contribution is a genericity theorem showing that the required rank/dimension conditions hold for a broad class of inputs, making the algorithms widely applicable. The results have potential impact across algebraic complexity, numerical cubature, and quantum dynamics by enabling reliable construction and classification of commuting extensions. The paper also discusses existence and non-uniqueness phenomena, and outlines directions for extending the theory to the two-matrix case and beyond .

Abstract

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.
Paper Structure (18 sections, 20 theorems, 18 equations)

This paper contains 18 sections, 20 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Our results
  3. Methods
  4. Where to look for a commuting extension
  5. Further work
  6. Organization of the paper
  7. Existence and non-uniqueness of commuting extensions
  8. Uniqueness Theorems
  9. Commutators
  10. Uniqueness for 3 matrices: proof of Theorem \ref{['th:3unique']}
  11. Uniqueness for more than 3 matrices
  12. Computing the extensions
  13. An algorithm for three matrices
  14. More than three matrices
  15. Some Generic Considerations
  16. ...and 3 more sections

Key Result

Theorem 1

Consider a tuple $(A_1,A_2,A_3)$ of matrices of size $n$ with entries in a field $K$ such that: The tuple $(A_1,A_2,A_3)$ does not have any commuting extension of size less than $r$. If $(A_1,A_2,A_3)$ has a commuting extension of size $r$, it is essentially unique. Moreover, if a commuting extension of size $r$ exists in the algebraic closure $\overline{K}$, there is already one in the ground

Theorems & Definitions (40)

  • Theorem 1
  • Example 2
  • Lemma 3
  • proof
  • Proposition 4
  • proof
  • Corollary 5
  • proof
  • Lemma 6
  • Theorem 7
  • ...and 30 more