*This question is a repost from math.se, where I didn't receive an answer.*

Are there simple conditions on an $d \times d$ matrix B under which
$$
f(t, s)
=
\begin{cases}
\exp(-B |t - s|^\alpha), & t > s, \\[7pt]
\exp(-B^\top |t - s|^\alpha), & t < s
\end{cases}
$$
with $\alpha \in (0,2)$ and $t, \, s \in \Bbb R$
is positive definite as a matrix-valued function *(see def below)*?

I know that it's true if $B$ is nonnegative definite (hence symmetric). But can these conditions be relaxed? Is it possible to show this for $B$ satisfying $B + B^\top \geq 0$ (eigenvalues have nonnegative real parts) without $B$ being symmetric (in particular, $B$ needs not be diagonalizable)?

**Example.** Consider a standard normal vector $\mathbf{N}$ and an antisymmetric matrix $V$. Define a Gaussian process $\mathbf{X} (t) = \exp(t V) \, \mathbf{N}$. Then
$$
\mathbb{E} \mathbf{X} (t) \, \mathbf{X}^\top (s) = \exp(V (t — s))
= \begin{cases}
\exp(V |t — s|), & t > s, \\[7pt]
\exp(V^\top |t — s|), & t < s
\end{cases}
$$
hence it is positive definite.

Is it true in general that if $r(t, s)$ is a positive definite **matrix-valued** function, then so is $\exp(r(t, s))$? This is obviously true if $r$ is scalar valued.

The reason for this particular question is that I am interested in the corresponding class of multivariate fractional Ornstein-Uhlenbeck processes.

P.S. By positive definiteness I mean $$ \sum_{i, j} \mathbf{x}_i^\top f (t_i, t_j) \, \mathbf{x}_j \geq 0 \qquad \forall \, t_i \in \mathbb{R}, \quad \mathbf{x}_i \in \mathbb{R}^d. $$