Abstract
The inertia of real matrices of the form C=[ADIO] and of the sign pattern matrix C of C are considered, where A is irreducible and D is a positive diagonal matrix. Such matrices C occur in applications that give rise to systems of linear second order differential equations. If D=dI is a positive scalar matrix, then the eigenvalues of C are determined by the magnitude of d and the eigenvalues of A, giving results for the inertia of C when A has certain specified forms. Results for the inertia of C are also given when D is an arbitrary positive diagonal matrix. The set of possible refined inertias of general sign pattern matrices C of the above form are determined, and are completely specified when A is a spectrally arbitrary sign pattern of order 2 or 3.
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