Hermitian operators nondegenerate

The solutions for the unperturbed Hamilton operator from a complete set (since Ho is hermitian) which can be chosen to be orthonormal, and A is a (variable) parameter determining the strength of the perturbation. At present we will only consider cases where the perturbation is time-independent, and the reference wave function is nondegenerate. To keep the notation simple, we will furthermore only consider the lowest energy state. The perturbed Schrodinger equation is... 

We first require Theorem 1. Let X be an operator defined in a finite space. X is Hermitian iff (1) its eigenvalues are real, and (2) its nondegenerate eigenvectors are mutually orthogonal. It is well known [151] that if X is Hermitian then (1) and (2) follow, and that neither (1) nor (2) is a sufficient condition for the Hermiticity of X. However, properties (1) and (2) together imply that X is Hermitian [152]. 

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