Journal of Applied Mathematics
Volume 2013 (2013), Article ID 597628, 13 pages
http://dx.doi.org/10.1155/2013/597628
Research Article

Neural-Network-Based Approach for Extracting Eigenvectors and Eigenvalues of Real Normal Matrices and Some Extension to Real Matrices

Xiongfei Zou,1 Ying Tang,2 Shirong Bu,1 Zhengxiang Luo,1 and Shouming Zhong3

1School of Optoelectronic Information, University of Electronic Science and Technology of China, Chengdu 610054, China
2School of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China
3School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China

Received 30 October 2012; Accepted 16 January 2013

Academic Editor: Nicola Mastronardi

Copyright © 2013 Xiongfei Zou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper introduces a novel neural-network-based approach for extracting some eigenpairs of real normal matrices of order n. Based on the proposed algorithm, the eigenvalues that have the largest and smallest modulus, real parts, or absolute values of imaginary parts can be extracted, respectively, as well as the corresponding eigenvectors. Although the ordinary differential equation on which our proposed algorithm is built is only n-dimensional, it can succeed to extract n-dimensional complex eigenvectors that are indeed 2n-dimensional real vectors. Moreover, we show that extracting eigen-pairs of general real matrices can be reduced to those of real normal matrices by employing the norm-reducing skill. Numerical experiments verified the computational capability of the proposed algorithm.