# The Second Order Directional Derivative of Symmetric Matrix-valued Functions

This paper focuses on the study of the second-order directional derivative of a symmetric matrix-valued function of the form $F(X)=P\mbox{diag}[f(\lambda_1(X)),\cdots,f(\lambda_n(X))]P^T$. For this purpose, we first adopt a direct way to derive the formula for the second-order directional derivative of any eigenvalue of a matrix in Torki \cite{Tor01}; Second, we establish a formula for the (parabolic) second-order directional derivative of the symmetric matrix-valued function. Finally, as an application, the second-order derivative for the projection operator over the SDP cone is used to derive the formula for the second-order tangent set of the SDP cone in Bonnans and Shapiro \cite{BS00}, which is the key for the Sigma term in the second-order optimality conditions of nonlinear SDP problems.