| Abstract: | It is well-known that if we have an approximate eigenvalue $\widehat{\lambda}$ 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 $x_k,k=1,\cdots,n$, obtained by simple inverse iteration, i.e., the solution to the system $(A-\widehat{\lambda}I)x=e_k$ with $e_k$ the $k$th 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. A stable double orthogonal factorization method and a simpler but may less stable approach are proposed for locating the largest component of $u$. |
