(untitled) (Journal of Computational Mathematics, 2021)
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 k_max, of the largest component of u is known. In this paper we give a detailed theoretical analysis to show relations between the eigenvector u and vector x_k,k=1,…,n, obtained by simple inverse iteration, i.e., the solution to the system (A-λI)x=e_k with e_k the kth column of the identity matrix I. We prove that under some weak conditions, the index k_max is of some optimal properties related to the smallest residual and smallest approximation error to u in spectral norm and Froenius norm. We also prove that the normalized absolute vector v=|u|/\|u\|_∞ of u can be approximated by the normalized vector of (\|x_1\|_2,…,\|x_n\|_2)^T. We also give some upper bounds of |u(k)| for those "optimal" indexes such as Fernando's heuristic for k_max 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.