Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
117 tokens/sec
GPT-4o
8 tokens/sec
Gemini 2.5 Pro Pro
47 tokens/sec
o3 Pro
5 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

On the uniqueness and computation of commuting extensions (2401.01302v1)

Published 2 Jan 2024 in cs.DS, cs.CC, and math.RA

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.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (13)
  1. Exponential rise of dynamical complexity in quantum computing through projections. Nature Communications, 5, 2014.
  2. Commuting extensions and cubature formulae. Numerische Mathematik, 101:479–500, 2005.
  3. Calculating multidimensional discrete variable representations from cubature formulas. The Journal of Physical Chemistry A, 110(16):5395–5410, 2006.
  4. Robert M Guralnick and BA Sethuraman. Commuting pairs and triples of matrices and related varieties. Linear algebra and its applications, 310(1-3):139–148, 2000.
  5. Components and singularities of Quot schemes and varieties of commuting matrices. Journal für die reine und angewandte Mathematik (Crelles Journal), 2022(788):129–187, 2022.
  6. Matrices with normal defect one. arXiv preprint arXiv:0903.0090, 2009.
  7. Minimal normal and commuting completions. International Journal of Information and Systems Sciences, 4(1):50–59, 2008.
  8. Pascal Koiran. On tensor rank and commuting matrices. arXiv preprint arXiv:2006.02374, 2020.
  9. Absolute reconstruction for sums of powers of linear forms: degree 3 and beyond. Computational Complexity, 32(2):1–66, 2023.
  10. Pairs of matrices with property L𝐿Litalic_L. II. Transactions of the American Mathematical Society, 80(2):387–401, 1955.
  11. Advanced topics in linear algebra: weaving matrix problems through the Weyr form. Oxford University Press, 2011.
  12. Hamiltonian purification. Journal of Mathematical Physics, 56(12), 2015.
  13. Volker Strassen. Rank and optimal computation of generic tensors. Linear Algebra and its Applications, 52:645–685, 1983.
Citations (2)

Summary

We haven't generated a summary for this paper yet.