This investigation demonstrates conditions for positive semidefinite solutions in LMIs, highlighting connections to stability theory and network dynamics.
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. (2025) studied this question.
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