2025/08/27 by Pazzis, Clément de Seguins
#15A18 #15A23 #15B33 #FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2508.19627
Let Q be a quaternion division algebra over a field, and n ≥ 2 be an integer. In a recent article, de La Cruz et al have proved that every n-by-n matrix with entries in Q and pure quaternionic trace is the sum of three nilpotent matrices, and they have shown that some are not the sum of two nilpotent matrices. Here, we give a simple characterization of the square matrices with entries in Q that are the sum of two nilpotent ones. When n ≥ 3, the special cases involve the scalar matrices and their perturbations by rank 1 matrices, as well as the very special case of 3-by-3 unispectral diagonalisable matrices.