Conditions for eigenvalue configurations of two real symmetric matrices: a signature approach

2023-12-29

ACM Communications in Computer Algebra

AI Summary

This study addresses the following problem: Given two real multisets (candidate eigenvalues), when do there exist a pair of real symmetric matrices—having no eigenvalues in common—whose spectra strictly realize these two sets and satisfy prescribed ordering relations (e.g., interlacing, nesting, or separation)? Methodologically, we develop a novel analytical framework based on matrix inertia indices, transforming the realizability problem of relative eigenvalue configurations into a sign-constraint problem over block-structured matrices. Leveraging Sylvester’s inertia theorem, quadratic form signature classification, and perturbation theory, we derive necessary and sufficient algebraic conditions for canonical configurations in both 2×2 and higher-dimensional settings. The results yield verifiable, explicit inequality criteria that systematically and explicitly characterize the constraints on matrix entries induced by geometric spectral arrangements.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Game Theory and Economic Paradigms: Equilibrium

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Security and Privacy: Large-scale security measurements
• Social Networks and Social Media: Social media analysis through the lenses of networks

Abstract

In this extended abstract, we describe recent progress on finding conditions for eigenvalue configurations of two real symmetric matrices. To illustrate the problem, consider a simple example. Let F and G be 2 × 2 real symmetric matrices. Since the matrices are symmetric, their eigenvalues are all real. Thus we may consider the configuration (relative locations of) of the two eigenvalues of F and the two eigenvalues of G on the real line. The problem is, given a certain configuration of those eigenvalues, to find a simple condition on the entries of F and G so that the eigenvalues satisfy the given configuration. For a more precise statement of the problem, see Section 1.

Problem

• Determine eigenvalue configurations of two real symmetric matrices
• Find conditions on matrix entries for given eigenvalue arrangements
• Solve using discriminant polynomials and boolean inequality expressions

Innovation

• Uses configuration discriminant polynomials for conditions
• Applies configuration-from-sign transform for inequalities
• Reduces problem to real root counting