The Field of Values of a Matrix and Neural Networks

The field of values of a matrix, also known as the numerical range, is introduced in the context of neural networks. Using neural network techniques, an algorithm and a generalization are developed that find eigenpairs of a normal matrix. The dynamics of the algorithm can be observed on the complex plane. Only limited visualization is possible in the case when the matrix is Hermitian (or real symmetric) since the field of values is confined on the real line. The eigenpairs can serve as stored memories, which are recalled by using the algorithm. Shifting in the algorithm is also discussed, which assists in finding other eigenpairs. Trajectories of runs of the algorithm are visually presented, through which the behavior of the algorithms is elucidated.