Balanced model reduction is a technique for producing a low dimensional approximation to a linear time invariant system. An important feature of balanced reduction is the existence of an error bound that is closely related to the decay rate of the eigenvalues of certain system Gramians. Rapidly decaying eigenvalues imply low dimensional reduced systems. New bounds are developed for the eigen-decay rate of the solution of Lyapunov equation AP + PA^T =- BB^T. These bounds take into account the low rank right hand side structure of the Lyapunov equation. They are valid for any diagonalizable matrix A. Numerical results are presented to illustrate the effectiveness of these bounds when the eigensystem of A is moderately conditioned.

We also present a bound on the norm of the solution P when A is diagonalizable and derive bounds on the conditioning of the Lyapunov operator for general A.

Key words: Lyapunov equation; Lyapunov operator; Low rank; Conditioning; Eigenvalue decay rate.