Further results on the measurement of solution bounds of the generalized Lyapunov equations. (Q1428207)

Generalized Lyapunov equations of the following type are studied: \[ d_0P+ d_1(A^* P+ PA)+ d_2 A^*PA+ d_3((A^*)^2 P+ PA^2)= -Q. \] Here \(A\) and \(Q\) are given matrices (with complex entries) and \(P\) is to be found. The matrices \(Q\) and \(P\) are assumed to be positive definite. Various upper and lower bounds for \(P\) and for the largest and the smallest eigenvalues of \(P\) are derived using iterative procedures and linear algebraic techniques, such as well-known inequalities between functions of eigenvalues of Hermitian matrices.