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 (λi), 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 λi 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.