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September 10, 2025Open Access

A Characterization of Diagonal Solutions for a Class of Linear Matrix Inequality

Key Points

Positive diagonal solutions for a class of linear matrix inequalities were identified, expanding existing theories.
Key results include equivalent conditions connecting negative definiteness with properties of positive semidefinite matrices.
The approach involves linking structured block matrices to Hadamard transformations with implications for control theory.
These findings suggest potential advancements in stability analyses within network dynamics and beyond.

Abstract

This paper investigates the existence of positive diagonal solutions for a class of linear matrix inequalities (LMIs) involving a triple of real n n matrices (A₁, A₂, A₃). We provide equivalent conditions linking the negative definiteness of a structured block matrix to properties of positive semidefinite test matrices and P-matrices under Hadamard transformations. Our results extend classical stability theory, offering new characterizations with applications in control theory~Boyd1994, Gahinet1994 and network dynamics.

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Algefary et al. (Fri,) studied this question.

synapsesocial.com/papers/68c1a77a54b1d3bfb60e0bf8 https://doi.org/https://doi.org/10.29020/nybg.ejpam.v18i3.6417

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