A constructive method for the solution of the stability problem

Summary In this paper it is shown that the problem of solving the Liapounov matrix equationSM +M ^T S = –I is greatly simplified when the given real matrixM is in upper Hessenberg form. The solution is obtained as a linear combinationS = p _i S _i ofn linearly independent symmetric matricesS _i , whereS _i M +M ^T S _i =2D _i and p _i D _i = –1/2I. Explicit formulae are given for the elements of theS _i , andD _i while determination of thep _i requires the solution of ann ×n linear system.