Technology Category
Machine Learning: Matrix & Tensor MethodsConstraint Satisfaction and Optimization: Other Foundations of Constraint SatisfactionGame Theory and Economic Paradigms: Equilibrium
Conditions for eigenvalue configurations of two real symmetric matrices: a signature approach
🤖 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
Application Category
📝 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
Research questions and friction points this paper is trying to address.
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
Methods, ideas, or system contributions that make the work stand out.
Uses configuration discriminant polynomials for conditions
Applies configuration-from-sign transform for inequalities
Reduces problem to real root counting
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