Author(s) :
Raphael Loewy
,
Volker Mehrmann
Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 04-2008
MSC 2000
- 15A27 Commutativity
Abstract :
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.
Keywords :
quasi-commutative matrices, roots of unity, Potter's theorem