[Paper Review] Some Properties of 3x3 Octonionic Hermitian Matrices with Non-Real Eigenvalues

Tevian Dray, Jason Janesky+1 more|ArXiv.org|Oct 26, 2000

Advanced Mathematical Theories and Applications12 references3 citations

TL;DR

This paper investigates 3×3 octonionic Hermitian matrices with non-real eigenvalues, deriving a third-order characteristic equation but failing to solve it or generalize orthonormality results from the real eigenvalue case. The key contribution is the identification of structural anomalies—such as eigenvectors forming families of six rather than three—and conjectures on non-standard orthogonality and decomposition rules that may underlie a deeper algebraic framework for exceptional Jordan algebras in quantum physics.

ABSTRACT

We discuss our preliminary attempts to extend previous work on 2x2 Hermitian octonionic matrices with non-real eigenvalues to the 3x3 case.

Motivation & Objective

• To extend previous work on 2×2 octonionic Hermitian matrices with non-real eigenvalues to the 3×3 case.
• To investigate whether orthonormality and spectral decomposition theorems—valid for real eigenvalues—can be generalized to non-real eigenvalues in the 3×3 case.
• To explore the algebraic structure of the exceptional Jordan algebra (Albert algebra) via 3×3 octonionic Hermitian matrices with non-real eigenvalues.
• To examine the implications of non-associativity in octonionic matrix theory for quantum mechanical models, particularly in the context of superparticle and superstring theories.
• To conjecture a new framework for eigenvector families and orthogonality conditions that may resolve inconsistencies in standard decomposition approaches.

Proposed method

• Derives a 3rd-order characteristic equation for right eigenvalues of 3×3 octonionic Hermitian matrices using the Jordan product and Freudenthal determinant.
• Applies the Freudenthal product to define the determinant abstractly and computes it explicitly for matrices with real diagonal and octonionic off-diagonal entries.
• Uses the associator identity [a,b,c]d + a[b,c,d] = [ab,c,d] −[a,bc,d] + [a,b,cd] to analyze non-associative products in eigenvalue equations.
• Analyzes specific examples with non-real eigenvalues, including one where eigenvalues lie outside the complex subalgebra generated by the associator.
• Tests orthonormality and decomposition properties by constructing eigenvector sets and evaluating outer product sums.
• Proposes a generalized orthogonality condition (vλ v†)w = 0 involving the matrix A, suggesting a non-standard inner product structure.

Experimental results

Research questions

• RQ1 Can the characteristic equation for 3×3 octonionic Hermitian matrices with non-real eigenvalues be solved explicitly?
• RQ2 Do orthonormal eigenvector sets exist for non-real eigenvalues, and if not, what alternative orthogonality conditions govern the eigenspaces?
• RQ3 Why do eigenvectors appear in families of six rather than three in certain examples, and what does this imply for spectral decomposition?
• RQ4 Is the standard decomposition A = Σ λα vα vα† valid for non-real eigenvalues, or must it be reformulated?
• RQ5 Can a generalized decomposition rule be formulated that preserves associativity in the form (AU)U† = A(UU†) even when UU† ≠ I?

Key findings

• A 3rd-order characteristic equation for right eigenvalues of 3×3 octonionic Hermitian matrices is derived, but it remains unsolved in the non-real eigenvalue case.
• In Example 2, the sum of the outer products of six normalized eigenvectors equals twice the identity matrix, suggesting a hidden symmetry or structure.
• For matrices with non-real eigenvalues, eigenvectors do not form orthonormal triples in the standard sense, and the standard decomposition A = Σ λα vα vα† fails.
• The eigenvalues in Example 3 lie outside the complex subalgebra generated by the associator [a,b,c], indicating that non-real eigenvalues can span non-quaternionic subalgebras.
• The standard orthonormality condition (22) fails for Example 2, and no orthonormal triple containing w1 exists, ruling out decomposition in forms (21) or (49).
• The paper conjectures that eigenvectors may come in sets of six or more, and that a generalized decomposition rule involving (AU)U† = A(UU†) may hold even when UU† ≠ I.