Right eigenvalue equation in quaternionic quantum mechanics

We study the right eigenvalue equation for quaternionic and complex linear matrix operators defined in n-dimensional quaternionic vector spaces. For q uaternionic linear operators the eigenvalue spectrum consists of n complex values. For these operators we give a necessary and sufficient condition fo r the diagonalization of their quaternionic matrix representations. Our dis cussion is also extended to complex linear operators, whose spectrum is cha racterized by 2n complex eigenvalues. We show that a consistent analysis of the eigenvalue problem for complex linear operators requires the choice of a complex geometry in defining inner products. Finally, we introduce some examples of the left eigenvalue equations and highlight the main difficulti es in their solution.