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.
Algefary et al. (2025) studied this question.