Complex-valued neural networks for the Takagi vector of complex symmetric matrices

This paper proposes complex-valued neural network for computing the Takagi vectors corresponding to the largest Takagi value of complex symmetric matrices. We establish some properties of the complex-valued neural network. Based on the Takagi factorization of complex symmetric matrices, we establish an explicit representation for the solution of the neural network and analyze its convergence property. Under certain conditions, we design a strategy to computing the Takagi factorization of a complex symmetric matrix by the proposed neural network. As an application, we consider the left and right singular vectors associated with the largest singular value for complex Toeplitz matrices. We illustrate our theory via numerical examples.