An algebraic method for quaternion and complex Least Squares coneigen-problem in quantum mechanics

In the study of theory and numerical computations of quantum theory, in order to well understand the perturbation theory, experimental proposals and theoretical discussions underlying the quaternion and complex formulations, one meets problems of approximate solutions of quaternion and complex problems, such as approximate solution of quaternion and complex linear equations Aα∼≈αλ or Aα‾≈αλ that is appropriate when there are errors in the vector α and λ, i.e. quaternion and complex Least Squares coneigen-problem in quantum mechanics. This paper, by means of representation of quaternion matrices and complex matrices, studies the problems of quaternion and complex Least Squares coneigen-problem, and give practical algebraic methods of computing approximate coneigenvalues and coneigenvectors for quaternion and complex matrices in quantum mechanics.