Let $S\subset\mathbb{R}$ be a convex set and $\mathbb{S}^{n}$ be the set of real symmetric matrices of order $n\times n$.

A matrix valued function $\Gamma: S \rightarrow \mathbb{S}^{n}$ is said to be convex if for all $x_1,x_2 \in S$ and for all $\lambda \in (0,1)$ one has $$\Gamma\left(\lambda x_{1}+\left(1-\lambda\right)x_{2}\right)\preceq\lambda\Gamma\left(x_{1}\right)+\left(1-\lambda\right)\Gamma\left(x_{2}\right),$$ where $\prec$ denotes the Loewner partial order, i.e. $A \prec B$ if $A - B$ is negative definite.

Is the matrix valued function $f: [0,c] \rightarrow \mathbb{S}^{n}$ given by $$ f(t) = e^{At} e^{A^{T} t} $$ convex for any $A \in \mathbb{R}^{n\times n}$ and $c>0$?

Numerical experiments suggest that this statement is true, but I could not prove.

Mike Juryhas an excellent explicit counterexample), a much weaker related claim is true.Claim.Let $A \in M_n(\mathbb{C})$. The function $f: [0,c] \to \mathbb{R}_+$ defined by \begin{equation*} f(t) = \|e^{tA}\| \end{equation*} is convex for any unitarily invariant norm $\|\cdot\|$. $\endgroup$ – Suvrit May 13 '15 at 18:011more comment