Explicit solution and eigenvalue bounds in the Lyapunov matrix equation

Abstract

An explicit solution to the algebraic Lyapunov matrix equation is obtained in terms of the controllability matrix of the pair of coefficient matrices. This enables us to determine the number of positive eigenvalues of the positive semidefinite solution through the controllability matrix. Based on this explicit formula, upper and lower bounds for each eigenvalue of the solution are derived, which always give nontrivial estimates.