Study On Directional Derivative Of Maximum Z-eigenvalue Function Of Real Symmetric Tensor

Tensor is an extension of matrix,which can be used to express and characterize very complex mathematical problems,especially closely related to polynomial optimization.This paper focuses on the differential properties of eigenvalues of tensors,and mainly studies the calculation formulas of directional derivatives for the maximum Z-eigenvalue function of even order real symmetric tensors and m order n dimensional real diagonal symmetric tensors.The Danskin theorem is applied to obtain the first-order directional derivative of the eigenvalue function Zthe maximum real symmetric tensor.Based on this,as an application,the first-order directional derivative of Z-the maximum eigenvalue function of the 4th order 2-dimensional real diagonal tensor and real sparse symmetric tensor is discussed.Lastly,a conclusion is given about the directional derivative of n real diagonal symmetric tensor of any m order.