Improved bounds for the eigenvalues of solutions of Lyapunov equations

Bounds on the solution and on the eigenvalues of the solution of the discrete and of the continuous matrix Lyapunov equations are presented. These bounds are nontrivial in the case of system matrices having multiple eigenvalues. Especially in the latter case, these bounds are better than bounds found in the literature. Further, these bounds are sharp, i.e., there exist systems such that the lower and upper bounds coincide yielding the solution or its eigenvalues, respectively.