# Is there a definition for a matrix Gaussian process?

Consider this definition of a Gaussian Process:

For any set $$π$$, a Gaussian Process ($$\mathcal{GP}$$) on $$π$$ is a set of random variables $$\{π_π‘:π‘\in π\}$$ such that $$\forallπ \in \mathbb{N}$$, $$\forall t_1,...,π‘_π \in S$$, $$\{π_{π‘_1},...,π_{π‘_π}\}$$ 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?

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