Solution of the Lyapunov matrix equation with a diagonal input matrix, obtained without matrix inversion

The solution of the Lyapunov matrix equation A¿L+LA=¿K with K=diag (¿i), i=1, 2,¿, n, is obtained via the Schwarz canonical form, by a method which requires no matrix inversion. The matrixes involved are formed by simple recursive schemes. A restriction on validity is that, corresponding to each nonzero ¿i and the ndimensional row vector y¿, the only nonzero element of which is unity in the ith position, the n×n matrix Ni, the rows of which, taken in sequence, are y¿, y¿A, ¿, y¿An¿1, must be nonsingular. At the cost of a matrix inversion, the scheme may be extended to the case that K is any symmetric real matrix.