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Neural-network-based approach for extracting eigenvectors and eigenvalues of real normal matrices and some extension to real matrices. (English) Zbl 1266.65062
Summary: 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 eigenpairs 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.
MSC:
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
92B20 Neural networks for/in biological studies, artificial life and related topics
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