Solution of some algebraic problems arising in the theory of stability and sensitivity of systems, with particular reference to the Lyapunov matrix equation

The matrix equation A'P + PA = -Q arises when the direct method of Lyapunov is used to analyse the stability of a constant linear system of differential equations ẋ = Ax. Considerable attention is given to the solution of this equation for the symmetric matrix P, given a symmetric positive definite matrix Q. Several new methods are proposed, including a reduction in the number of equations and unknowns brought about by introducing a skew-symmetric matrix; a method based on putting A into Schwarz form and inverting a triangular matrix; and a solution in terms of a convergent infinite matrix series. Some numerical experience is also reported. [Continues.