COMPUTING EIGENVECTORS OF NORMAL MATRICES WITH SIMPLE INVERSE ITERATION
Zhen-yueZhang, Tiang-weiQuyang · Acta Scientiarum Naturalium Universitatis Sunyatseni · 2003
It is well-known that if we have an approximate eigenvalue λ- of a normal matrix A of order n,a good approximation to the corresponding eigenvector u can be computed by one inverse iteration provided the position,say kmax,of the largest component of u is known.In this paper we give a detailed theoretical analysis to show relations between the eigenvecor u and vector xk,k=1,…,n,obtained by simple inverse iteration,i.e.,the solution to the system(A-λI)x=ek with ek the kth column of the identity matrix I.We prove that under some weak conditions,the index kmax is of some properties related to the smallest residual and smallest approximation error to u in spectral norm and Frobenius norm.We also prove that the normalized absolute vector v=|u|/||u||∞ of u can be approximated by the normalized vector of (||x1||2,…||xn||2)^T，We also give some upper bounds of |u(k)| for those optimal indexeds such as Fernando's heuristic for kmax without any assumptions,A stable double orthogonal factorization method and a simpler but may less stable approach are proposed for locating the largest component of u.