Let $p$ be a positive integer and suppose that each observation in my data set is a length-$p$ multivariate normal vector, and I have $n$ (an integer) observations of the length-$p$ multivariate normal vector. So $$ \vec{Y} = \beta_0 + \beta_1 \vec{X}_{1} + \cdots + \beta_k \vec{X}_{k} + \vec{\epsilon}, $$ with $\vec{\epsilon} \sim N_p(\vec{0}, \Sigma) $, $\Sigma$ is a covariance matrix of an observation-vector, $\beta_i \in \mathbb{R}$ (for $i \in \{0,1,\cdots,k\}$) and $X_i \in \mathbb{R}^p$. I am in a situation where this model looks relevant to my problem, but I have never been taught how to generalize the usual regression model into one where each observation is itself a vector of size $p>1$.
Is this called multivariate multiple regression? How can I find literature for it? If I look up multivariate-, or multidimensional linear regression I only get stuff on the multivariate linear regression model (the case where $p=1$).
