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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 positivedefinite matrices $A$. Under certain continuity conditions on~$f$, thisfunction may be extended to certain non-positive-definite matrices $A$, and the\emph{directional} derivatives of $\Tr(Pf(A)$ may be computed there. This notepresents conditions for these directional derivatives to exist and computesthem. These conditions hold for the function $f(x)=\log(x)$ and for thefunctions $f_p(x)=x^p$ for all $p>-1$. The derivatives of the correspondingtrace functionals are computed here.

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