A Faddeev sequence method for solving Lyapunov and Sylvester equations

A two-variable approach to solve the polynomial Lyapunov equation

“…It follows by the same arguments as used in the proof of [6,Lemma 5:1], that the trace of an operator M of the form M : Y → PYQ T + QYP T , acting on n × n constant symmetric matrices, and with P and Q real matrices of size n × n, is given by trace(M ) = trace(P)trace(Q) + trace(PQ). Hence, the trace u k of L k R is given by for k = 0; 1; : : : ; d. Note that trace(C k ) = t k .…”

“…Therefore, it only remains to prove that the symmetric functions t k are related to the symmetric functions u k by means of recursion (5.9). This is achieved by considering properties of the Faddeev algorithm of [6,Section 5] for the symmetric matrix Lyapunov equation.…”

“…Second, the actual algorithm for solving the PLE proceeds by solving the LPLE by an iterative method which is inspired by the Faddeev sequence-based approach of [6] to solve matrix Sylvester and Lyapunov equations. Thus designed, the method turns out to be especially suited for applications requiring symbolic or exact computation [9].…”

Reachability matrices and cyclic matrices

“…We refer to [8] for an investigation of numerical aspects of Krylov and reachability matrices. The concept of Faddeev reachability matrix was introduced in [5] and further elaborated in [10]. A "spectral factorization" of R(A, b) is due to [7] (see also [13]).…”

A two-variable approach to solve the polynomial Lyapunov equation

“…It follows by the same arguments as used in the proof of [6,Lemma 5:1], that the trace of an operator M of the form M : Y → PYQ T + QYP T , acting on n × n constant symmetric matrices, and with P and Q real matrices of size n × n, is given by trace(M ) = trace(P)trace(Q) + trace(PQ). Hence, the trace u k of L for k = 0; 1; : : : ; d. Note that trace(C k ) = t k .…”