On directional derivatives of trace functionals of the form $A\mapsto\Tr(Pf(A))$

TL;DR

This paper investigates the existence and computation of directional derivatives of trace functionals of the form $A o ext{Tr}(Pf(A))$, extending their definition beyond positive definite matrices for functions like log and power functions.

Contribution

It establishes conditions for the existence of directional derivatives of trace functionals and explicitly computes these derivatives for key functions such as log and power functions.

Findings

Derived conditions for directional derivatives to exist.

Computed derivatives for $f(x)= ext{log}(x)$ and $f_p(x)=x^p$.

Extended the trace functional to certain non-positive-definite matrices.

Abstract

Given a function $f:(0,\infty)\rightarrow\RR$ and a positive semidefinite $n\times n$ matrix $P$, one may define a trace functional on positive definite $n\times n$ matrices as $A\mapsto \Tr(Pf(A))$. For differentiable functions $f$, the function $A\mapsto \Tr(Pf(A))$ is differentiable at all positive definite matrices $A$. Under certain continuity conditions on~$f$, this function may be extended to certain non-positive-definite matrices $A$, and the \emph{directional} derivatives of $\Tr(Pf(A)$ may be computed there. This note presents conditions for these directional derivatives to exist and computes them. These conditions hold for the function $f(x)=\log(x)$ and for the functions $f_p(x)=x^p$ for all $p>-1$. The derivatives of the corresponding trace functionals are computed here.