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.]
A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy at Loughborough University.