Volume 21, Issue 5
Computing Eigenvectors of Normal Matrices with Simple Inverse Iteration

Zhen-yue Zhang & Tiang-wei Ouyang

DOI:

J. Comp. Math., 21 (2003), pp. 657-670

Published online: 2003-10

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  • 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

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@Article{JCM-21-657, author = {Zhen-yue Zhang and Tiang-wei Ouyang}, title = {Computing Eigenvectors of Normal Matrices with Simple Inverse Iteration}, journal = {Journal of Computational Mathematics}, year = {2003}, volume = {21}, number = {5}, pages = {657--670}, 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. }, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/10244.html} }
TY - JOUR T1 - Computing Eigenvectors of Normal Matrices with Simple Inverse Iteration AU - Zhen-yue Zhang & Tiang-wei Ouyang JO - Journal of Computational Mathematics VL - 5 SP - 657 EP - 670 PY - 2003 DA - 2003/10 SN - 21 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/10244.html KW - Eigenvector KW - Inverse iteration KW - Accuracy KW - Error estimation AB - 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.
Zhen-yue Zhang & Tiang-wei Ouyang. (1970). Computing Eigenvectors of Normal Matrices with Simple Inverse Iteration. Journal of Computational Mathematics. 21 (5). 657-670. doi:
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