Some new inertia theorems for the discrete-time algebraic Lyapunov equation

This paper deals with the properties of the solutions of the discrete-time algebraic Lyapunov equation (DALE) under assumptions of reconstructibility or detectability of the underlying system. The inertia (i.e. the number of positive, null and negative eigenvalues) of any symmetric solution is linked with the inertia of the dynamic matrix and the dimension of the unobservability subspace. Some corollaries for the case of positive semidefinite solutions are also provided.