I am interested in homogeneity degree one (scalar-valued) functions of a matrix argument. The simplest setup is as follows. Let $X$ be a symmetric $3\times 3$ matrix with real entries. Let $f$ be a real-valued function $f(X)\in{\mathbb R}$ that takes the same value on the matrices related by an orthogonal transformation, i.e. $f(O X O^T)=f(X), \forall O\in {\rm SO}(3)$. In addition, let $f$ be a function of homogeneity degree one, i.e. $f(\alpha X)=\alpha f(X)$. As such, these functions can be thought of as (symmetric) homogeneity degree one functions of the eigenvalues $X=O \,{\rm diag}(\lambda_1,\lambda_2,\lambda_3) O^T$, $f(X)=f(\lambda_1,\lambda_2,\lambda_3)$, or as functions on (a quotient of) the real projective space ${\mathbb RP}^2$ with $\lambda_1,\lambda_2,\lambda_3$ being the homogeneous coordinates. However, this way of thinking about these functions is not very well suited for practical manipulations, e.g. for the problem of computing the matrices of partial derivatives $\partial f^k/\partial X\partial X \ldots \partial X$.
Question 1: Is there some representation (e.g. some sort of integral transform) of such functions that makes computations of partial derivatives easy?
Question 2: What is special (e.g. as an object in ${\mathbb RP}^2$) about the following function (this time defined only for definite matrices): $$f(X) = \left( \sqrt{\lambda_1}+\sqrt{\lambda_2}+\sqrt{\lambda_3}\right)^2$$
