On families of anticommuting matrices

TL;DR

This paper investigates the properties of families of complex matrices that anticommute, proposing a conjecture relating their ranks and providing partial results towards this conjecture.

Contribution

The paper introduces a conjecture on the sum of ranks of squared matrices in anticommuting families and proves partial results supporting this conjecture.

Findings

Proposed a conjecture relating ranks and matrix size.

Established bounds and partial results towards the conjecture.

Contributed to understanding the structure of anticommuting matrices.

Abstract

Let $e_{1},\dots, e_{k}$ be complex $n\times n$ matrices such that $e_{i}e_{j}=-e_{j}e_{i}$ whenever $i\not=j$. We conjecture that $\hbox{rk}(e_{1}^{2})+\hbox{rk}(e_{2}^{2})+\cdots+\hbox{rk}(e_{k}^{2})\leq O(n\log n)$, and prove some results in this direction.