A Characterization of Diagonal Solutions for a Class of Linear Matrix Inequality (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~\cite{Boyd1994,Gahinet1994} and network dynamics.
Keywords: diagonal matrices; $P$-matrix; positive definite matrix; matrix inequality; matrix stability