Right eigenvalue equation in quaternionic quantum mechanics

TL;DR

This paper investigates the right eigenvalue problem in quaternionic quantum mechanics, characterizing spectra of quaternionic and complex linear operators and discussing diagonalization conditions and eigenvalue complexities.

Contribution

It provides a comprehensive analysis of right eigenvalues for quaternionic and complex linear operators, including diagonalization criteria and spectrum characterization.

Findings

Quaternionic operators have n complex eigenvalues.

Complex linear operators have 2n complex eigenvalues.

Diagonalization conditions depend on quaternionic matrix properties.

Abstract

We study the right eigenvalue equation for quaternionic and complex linear matrix operators defined in n-dimensional quaternionic vector spaces. For quaternionic linear operators the eigenvalue spectrum consists of n complex values. For these operators we give a necessary and sufficient condition for the diagonalization of their quaternionic matrix representations. Our discussion is also extended to complex linear operators, whose spectrum is characterized 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 difficulties in their solution.