(untitled) (European Journal of Pure and Applied Mathematics, 2025)
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 (A_1, A_2, A_3). 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.