Eigenvectors in terms of reduced complements of minor determinants

TL;DR

This paper explores the relationship between eigenvectors and determinants of the characteristic matrix, providing new insights into degenerate and non-degenerate cases with applications to quantum systems.

Contribution

It introduces a novel tensor-based approach to eigenvector characterization using reduced complements of minor determinants, extending classical eigenvector theory.

Findings

Eigenvectors for non-degenerate eigenvalues relate to the adjugate matrix.

Degenerate eigenvalues correspond to reduced complement tensors.

Trace identities for these tensors are established.

Abstract

Eigenvectors associated with non-degenerate eigenvalues are shown to correspond to columns of the adjugate of the characteristic matrix. Degenerate eigenvalues are associated with eigenvectors that correspond to reduced complement tensors of minor determinants of the characteristic matrix. These observations are corroborated by a description of the non-degenerate two-level system and the Dirac equation, which exhibits twofold spin degeneracy of energy eigenvalues. Trace identities for the reduced order-one complement tensor and the diagonal sum of minor determinants are also presented.