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arXiv · 2608.27283

The Matrix Pythagorean Equation over $\mathrm{GL}_2(\mathbb{Z})$

Hongjian Li·Weilin Zhang
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Abstract

In this paper, we study the ordered solutions of the matrix Pythagorean equation $X^2+Y^2=Z^2$ over $\mathrm{GL}_2(\mathbb{Z})$ . Exploiting the independent sign symmetry of the equation, we first reduce the full solution set to the trace-nonnegative subset $\mathcal M$, consisting of those solutions for which all three traces are nonnegative. We then determine a canonical decomposition of $\mathcal M$ into orbits under simultaneous integral conjugacy.

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Hongjian Li, Weilin Zhang. 2026-08-27. The Matrix Pythagorean Equation over $\mathrm{GL}_2(\mathbb{Z})$. https://arxiv.org/abs/2608.27283

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