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Eigenvectors in terms of reduced complements of minor determinants

M. I. Krivoruchenko

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.

Eigenvectors in terms of reduced complements of minor determinants

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.
Paper Structure (9 sections, 11 theorems, 83 equations)

This paper contains 9 sections, 11 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Basic notations and definitions
  3. Eigenvectors for non-degenerate eigenvalues
  4. Eigenvectors for degenerate eigenvalues
  5. Basic properties of $\psi_{i_{1}\mathfrak{\ldots} i_{r}}^{j_{1}\mathfrak{\ldots} j_{r}} (s,\lambda)$
  6. Product of $A_{i}^{j}$ and $(\underset{s}{\det }\, \mathbf{A})_{i_{1}\mathfrak{\ldots }i_{r}}^{j_{1}\mathfrak{\ldots }j_{r}}$
  7. Basic properties of $\psi_{i}^{j} (s,\lambda)$
  8. Trace identity for $\psi_{i}^{j} (s,\lambda)$
  9. Conclusions

Key Result

Theorem \oldthetheorem

Let $\mathbf{H}$ be an $n \times n$ Hermitian matrix and let $\lambda_{\mathfrak{d}}$ be an eigenvalue of $\mathbf{H}$ of algebraic multiplicity $\mu_{\mathbf{H}}(\lambda_{\mathfrak{d}}) = 1$. In the diagonal coordinate system, the adjugate matrix $\psi_{\mathfrak{b}}^{\mathfrak{a}}(s = 1,\lambda_{\

Theorems & Definitions (28)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Remark 1
  • Theorem \oldthetheorem
  • Example 1
  • Theorem \oldthetheorem
  • Lemma 1
  • Corollary 1
  • ...and 18 more