Algebraic systems of matrices and Gröbner basis theory

The problem of finding all the n×n complex matrices A,B,C such that, for all real t, etA+etB+etC is a scalar matrix reduces to the study of a symmetric system (S) in the form: {A+B+C=αIn,A2+B2+C2=βIn,A3+B3+C3=γIn} where α,β,γ are given complex numbers. Except in a special case, we solve explicitly these systems, depending on the values of the parameters α,β,γ. For this purpose, we use Gröbner basis theory. A nilpotent algebra is associated to the special case. The method used for solving (S) leads to complete solution of the original problem. We study also similar systems over the n×n real matrices and over the skew-field of quaternions.