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

Xiongfei Zou,Ying Tang,Shouming Zhong,Shirong Bu,Zhengxiang Luo

doi:10.1155/2013/597628

Xiongfei Zou, Ying Tang + Show 3 more

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https://doi.org/10.1155/2013/597628

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Abstract

This paper introduces a novel neural-network-based approach for extracting some eigenpairs of real normal matrices of ordern. 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 onlyn-dimensional, it can succeed to extractn-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.

Highlights

The problem of extracting special eigenpairs of real matrices has attracted much attention both in theory [1,2,3,4] and in many engineering fields such as real-time signal processing [5,6,7,8] and principal or minor component analysis [9,10,11,12]
This paper introduces a novel neural-network-based approach for extracting some eigenpairs of real normal matrices of order n
We propose an approach for extracting six types of eigenvalues of n-by-n real normal matrices and the corresponding eigenvectors based on (2) or (3)

Summary

Introduction

The problem of extracting special eigenpairs of real matrices has attracted much attention both in theory [1,2,3,4] and in many engineering fields such as real-time signal processing [5,6,7,8] and principal or minor component analysis [9,10,11,12]. Many neuralnetwork-based methods have been proposed to solve this problem [14,15,16,17,18,19,20,21,22,23] Most of those neural network based methods focused on computing eigenpairs of real symmetric matrices. Were proposed by [19, 23], respectively, where A is a real symmetric matrix Both (1) and (2) are efficient to compute the largest eigenvalue of A, as well as the corresponding eigenvector. We propose an approach for extracting six types of eigenvalues of n-by-n real normal matrices and the corresponding eigenvectors based on (2) or (3). We show that any real matrix can be made arbitrarily close to a normal matrix by a series of similarity transformations, based on which our proposed algorithm can be extended to the case of arbitrary real matrices

Main Results

Computing the Eigenvalues with the Largest or Smallest

Neural Implementation Description

Examples and Discussion

Conclusion