Algebraic matrix equations in two unknowns

Details Algebraic matrix equations in two unknowns Algebraic matrix equations in two unknowns

2011-03-22

arxiv.org/abs/1103.4203v3

Mathematics

Rings and Algebras

10 pages

Scientific paper

Let p,q be coprime integers such that |p|+|q|>2. We characterize the n,n complex matrices A such that A^p and A^q are similar, that is essentially, we study the matrix equation B^{-1}A^pB=A^q where the n,n complex invertible matrices A,B are to be determined. We show that for such matrices B^{-1}AB and A commute. We explicitly solve this problem in the unknowns A,B when A has n distinct eigenvalues and in other particular cases. Finally the more general matrix equation A^rB^sA^r'B^s'=+- id, where r,r',s,s' are given integers such that gcd(r,r')=1 and gcd(s,s')=1 and where the 2,2 complex matrices A,B are to be determined, is solved.

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