A \(3\times 3\) counterexample to Lin and Wimmer's rank-minimization conjecture associated with Roth's similarity theorem

Shuo Shi; Juan Zhang; Yun Zhang

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arXiv · 2609.07113

A \(3\times 3\) counterexample to Lin and Wimmer's rank-minimization conjecture associated with Roth's similarity theorem

Shuo Shi·Juan Zhang·Yun Zhang

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Abstract

We give a \(3\times 3\) counterexample, valid over every field, to a rank-minimization conjecture of Lin and Wimmer (Bull. Aust. Math. Soc., 84 (3) (2011), 441--443) related to Roth's similarity theorem for the Sylvester matrix equation. We also prove that, over the complex field, no counterexample can occur when one of the two matrix sizes is less than \(3\). Hence the example is dimensionally minimal over the complex field.

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Shuo Shi, Juan Zhang, Yun Zhang. 2026-09-07. A \(3\times 3\) counterexample to Lin and Wimmer's rank-minimization conjecture associated with Roth's similarity theorem. https://arxiv.org/abs/2609.07113

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