• Abstract

It is well-known that if we have an approximate eigenvalue $\widehat{\lambda}$ of a normal matrix A of order n, a good qpproximation 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 $x_k,k=1,\cdots,n$, obtained by simple inverse iteration, i.e., the solution to the system $(A-\widehat{\lambda}I)x=e_k$ 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 $v=|u|/\|u\|_\infty$of u can be approximated by the normalized vector of $(\|x_1\|_2,\cdots,\|x_n\|_2)^T$. We also give some upper bounds of $|u(k)|$ for those "optimal" indexes such as Fernando;s heuristic for $k_{max}$ without any assumptions. Astable double orthogonal factorization method and a simpler but may less stable approach are proposed for locating the largest component of u.

• Keywords

Eigenvector Inverse iteration Accuracy Error estimation

• AMS Subject Headings

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