A constructive method for the solution of the stability problem

SummaryIn this paper it is shown that the problem of solving the Liapounov matrix equationSM +MTS = −I is greatly simplified when the given real matrixM is in upper Hessenberg form. The solution is obtained as a linear combinationS = ΣpiSi ofn linearly independent symmetric matricesSi, whereSiM +MTSi =2Di and ΣpiDi = −1/2I. Explicit formulae are given for the elements of theSi, andDi while determination of thepi requires the solution of ann ×n linear system.