On the Lyapunov matrix equation

In this paper the inequality which is satisfied by the determinant of the solution of the Lyapunov matrix equation
A'Q + QA = - D
is presented. The result makes possible a lower estimate of product eigenvalues of the matrix
Q
and dependence from eigenvalues of the matrices
A
and
D
. This result corresponds to those presented in [2] and [3], where an estimate of the extremal eigenvalues of the matrix
Q
is presented. This estimate depends on the eigenvalues of the matrices
A
and
D
.