A complex neural network algorithm for computing the largest sum of real part and imaginary part of eigenvalues and the corresponding eigenvector of a real normal matrix

In this study, we proposed a novel complex neural network algorithm, which extends the neural networks based approaches that can asymptotically compute the largest modulus of eigenvalues and the corresponding eigenvector to the case of directly computing the largest sum of real part and imaginary part of eigenvalues and the corresponding eigenvectors of a real normal matrix. The proposed neural network algorithm is described by a group of complex differential equations. And the algorithm has parallel processing ability in an asynchronous manner and could achieve high computing capability. This paper also provides a rigorous mathematical proof for its convergence for a more clear understanding of network dynamic behaviors relating to the computation of the eigenvector and the eigenvalue. Numerical example showed that the proposed algorithm has good performance for a general real normal matrix.