I have two random matrices one on the top of the other:

$ \begin{bmatrix}\boldsymbol{B_1} \\ \boldsymbol{B_2} \end{bmatrix}$.

and they are both of dimension $k \times N$. I have that:

$ vec\begin{bmatrix}\boldsymbol{B_1} \\ \boldsymbol{B_2} \end{bmatrix} \sim \mathcal{N}\left(\begin{bmatrix}\boldsymbol{\mu_1} \\ \boldsymbol{\mu_2}\end{bmatrix}, \boldsymbol{\Sigma} \otimes \boldsymbol{S} \right)$.

where $\boldsymbol{\Sigma}$ is an $N \times N$ matrix and $2k \times 2k$. I want to find the distribution of $vec(\boldsymbol{\boldsymbol{B}_2})| vec(\boldsymbol{B_1})$, how can I procede? It should be a normal, but how do I get the mean and the variance?



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