The Directional Derivative Of H-maximum Eigenvalue Function Of Real Symmetric Tensor

In this paper we mainly discuss the first order directional derivative of H-maximum eigen-value function of real symmetric tensor.The biggest difference between matrix case and tensor case is the eigenspace of matrix is linear space but H-eigenspace of tensor is not always linear space.So we give a general conclusion of H-maximum eigenvalue function of real symmetric tensor.On this basis we imitate the conclusion in matrix case and give a conclusion when the H-eigenspace of a sysmmetric tensor is linear.Then we find that H-eigenspace is linear space when the sysmmetric tensor is a diagonal tensor.Finally,we give a conclusion when the H-eigenspace is a nonlinear space,and this conclusion is based on the finite number of two linearly independent vectors in the H-eigenspace.