Is there a definition for a matrix Gaussian process?

Question

Consider this definition of a Gaussian Process:

For any set $$, a Gaussian Process ($\mathcal{GP}$) on $$ is a set of random variables $\{_:\in \}$ such that $\forall \in \mathbb{N}$, $\forall t_1,...,_ \in S$, $\{_{_1},...,_{_}\}$ has a multivariate Gaussian distribution.

What if each random variable $Z_t$ itself has a multivariate Gaussian distribution? Can we easily redefine a Gaussian process to say that "[..] the set $\{Z_{t_1}, ..., Z_{t_n}\}$ has a matrix Gaussian distribution"? Do we run into problems if we use this definition?

Answer 1

I found a definition in Sparse matrix-variate Gaussian process blockmodels for network modeling (Feng Yan, Zenglin Xu and Yuan Qi, 2012):

A matrix-variate Gaussian process is a stochastic process whose projection on any finite locations $U = [u^⊤_1, . . . , u^⊤_n]^⊤$ follows a matrix-variate normal distribution. Specically, given $U$, the zero mean matrix-variate Gaussian process on $M$ has the form: $p(M|U) =\mathcal{GP}_{n,n}(M; 0,K,G)= (2\pi)^{−n^2/2}\text{det}(K)^{−n/2} \text{det}(G)^{−n/2} \text{exp}\{−\frac{1}{2}\text{tr}(K^{−1}MG^{−1}M^⊤)\}$ where $k_{ij}=k(u_i,u_j)$ and $g_{ij}=g(u_i,u_j)$.

No one seems to talk much about matrix-variate Gaussian processes though.