In this paper, we examine the maximum eigenvalue function of an even order real symmetric tensor. By using the variational analysis techniques, we first show that the maximum eigenvalue function is a continuous and convex function on the symmetric tensor space. In particular, we obtain the convex subdifferential formula for the maximum eigenvalue function. Next, for an mth-order n-dimensional symmetric tensor A, we show that the maximum eigenvalue function is always ρth-order semismooth at A for some rational number ρ>0. In the special case when the geometric multiplicity is one, we show that ρ can be set as 1(2m-1)n. Sufficient condition ensuring the strong semismoothness of the maximum eigenvalue function is also provided. As an application, we propose a generalized Newton method to solve the space tensor conic linear programming problem which arises in medical imaging area. Local convergence rate of this method is established by using the semismooth property of the maximum eigenvalue function.
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