Stability of sign patterns from a system of second order ODEs

The stability and inertia of sign pattern matrices with entries in { + , − , 0 } associated with dynamical systems of second-order ordinary differential equations x ¨ = A x ˙ + B x are studied, where A and B are real matrices of order n. An equivalent system of first-order differential equations has coefficient matrix C = [ A B I O ] of order 2n, and eigenvalue properties are considered for sign patterns C = [ A B D O ] of order 2n, where A , B are the sign patterns of A , B respectively, and D is a positive diagonal sign pattern. For given sign patterns A and B where one of them is a negative diagonal sign pattern, results are determined concerning the potential stability and sign stability of C , as well as the refined inertia of C . Applications include the stability of such dynamical systems in which only the signs rather than the magnitudes of entries of the matrices A and B are known.