A functional neural network computing some eigenvalues and eigenvectors of a special real matrix

Abstract

How to quickly compute eigenvalues and eigenvectors of a matrix, especially, a general real matrix, is significant in engineering. Since neural network runs in asynchronous and concurrent manner, and can achieve high rapidity, this paper designs a concise functional neural network (FNN) to extract some eigenvalues and eigenvectors of a special real matrix. After equivalent transforming the FNN into a complex differential equation and obtaining the analytic solution, the convergence properties of the FNN are analyzed. If the eigenvalue whose imaginary part is nonzero and the largest of all eigenvalues is unique, the FNN will converge to the eigenvector corresponding to this special eigenvalue with general nonzero initial vector. If all eigenvalues are real numbers or there are more than one eigenvalue whose imaginary part equals the largest, the FNN will converge to zero point or fall into a cycle procedure. Comparing with other neural networks designed for the same domain, the restriction to matrix is very slack. At last, three examples are employed to illustrate the performance of the FNN.