Some applications of the Newton—Raphson method to non-linear matrix problems

If D(A) is a A-matrix the latent roots are defined as the roots, Ai, of the equation |D(A) | = 0, and the eigenvalues are defined as the roots, uj(A), of |D—uI| = 0. It is shown that uj(A) = 0 for some j if and only if A is a latent root, and it is shown th at the Newton—Itaphson method can be employed to find the zeros of the functions /qj(A) and hence the latent roots. In contrast to the solution of scalar equations, it is shown that, under certain conditions, the method is valid for multiple latent roots It is also shown that the method may be applied to stability problems of the kind arising in the study of aircraft flutter problems, and a numerical example is given.