Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Real valued functions, $F(X)$, on a symmetric matrix argument are called spectral if $F(U^TXU) = F(X)$ for every orthogonal matrix $U$ and $X \in \mathrm{dom\,} F$.  are interested in a description of the higher order derivatives (when they exist) of $F$ with respect to $X$. Formulae for the gradient and the Hessian of $F$ are given in [A. S. Lewis, Math. Oper. Res., 21 (1996), pp. 576-588] and [A. S. Lewis and H. S. Sendov, SIAM Matrix Anal. Appl., 23 (2001), pp. 368-386]. In this work  present common features of these two formulae that are preserved in the higher order derivatives.
['Hristo S. Sendov']
I am writing a research paper on a idea Generalized Hadamard Product and the Derivatives of Spectral Functions
[{"from": "human", "value": "I am writing a research paper on a idea Generalized Hadamard Product and the Derivatives of Spectral Functions"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Real valued functions, $F(X)$, on a symmetric matrix argument are called spectral if $F(U^TXU) = F(X)$ for every orthogonal matrix $U$ and $X \\in \\mathrm{dom\\,} F$.  are interested in a description of the higher order derivatives (when they exist) of $F$ with respect to $X$. Formulae for the gradient and the Hessian of $F$ are given in [A. S. Lewis, Math. Oper. Res., 21 (1996), pp. 576-588] and [A. S. Lewis and H. S. Sendov, SIAM Matrix Anal. Appl., 23 (2001), pp. 368-386]. In this work  present common features of these two formulae that are preserved in the higher order derivatives."}]