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A note on Potter's theorem for quasi-commutative matrices

Loewy, Raphael; Mehrmann, Volker

Preprint-Reihe des Instituts für Mathematik, Technische Universität Berlin

We discuss the converse of a theorem of Potter stating that if the matrix equation $AB = \omega BA$ is satisfied with $\omega$ a primitive $q$th root of unity, then $A^q + B^q = (A+B)^q$. We show that both conditions have to be modified to get a converse statement and we present a characterization when the converse holds for these modified conditions and $q=3$ and a conjecture for the general case. We also present some further partial results and conjectures.