Conjugate priors for dynamic model $x_{t+1}=Ax_{t}+\eta_t$ What conjugate priors do we have for the model(multivariate) $x_{t+1}=Ax_{t}+\eta_t$, where $\eta_t\overset{iid}\sim N(0,\Sigma)$?
I was thinking of using $\tilde{x}=Diag[x_1,...,x_{n-1}]$, $\tilde{y}=Diag[x_2,...,x_{n}]$ and simply use the usual conjugate priors for the Bayesian multivariate linear regression.
Will there be any problem, if I do this?
 A: Bottom Line
You're right. You can rewrite the model as a multivariate linear regression and simply use the conjugate prior for that model.
Some Detail
The model you wrote down is a vector autoregression. You wrote it with a single lag and no intercept term, but the general model with $p$ lags is
$$\underbrace{\mathbf{y}_t}_{n\times 1}=\underbrace{\mathbf{b}_0}_{n\times 1}+\underbrace{\mathbf{B}_1}_{n\times n}\mathbf{y}_{t-1}+...+\mathbf{B}_p\mathbf{y}_{t-p}+\underbrace{\mathbf{u}_t}_{n\times 1},\quad\mathbf{u}_t\sim N(\mathbf{0},\,\underbrace{\boldsymbol{\Sigma}}_{n\times n}).$$
If for each $t$ you define the vector
$$\underbrace{\mathbf{x}_t}_{(np+1)\times 1} = [\mathbf{y}'_{t-1} \quad \mathbf{y}'_{t-2} \quad...\quad \mathbf{y}'_{t-p}\quad 1]'$$
and the matrix
$$\underbrace{\mathbf{B}}_{(np+1)\times n}=\begin{bmatrix}\mathbf{B}'_1\\\mathbf{B}'_2\\\vdots\\\mathbf{B}'_p\\\mathbf{b}'_0\end{bmatrix}$$
then you can rewrite the VAR model as a multivariate regression:
$$\mathbf{y}'_t=\mathbf{x}'_t\mathbf{B}+\mathbf{u}'_t,\quad\mathbf{u}_t\sim N(\mathbf{0},\,\boldsymbol{\Sigma}).$$
The conjugate prior for $\mathbf{B}$ and $\mathbf{\Sigma}$ is the Matrix Normal Inverse Wishart Distribution. That is
$$\begin{align*}\mathbf{\Sigma}&\sim IW(d_0,\, \mathbf{\Psi}_0) \\
\mathbf{B}\,|\,\mathbf{\Sigma}&\sim MN(\bar{\mathbf{B}}_0,\, \mathbf{\Omega}^{-1}_0,\,\mathbf{\Sigma}).\end{align*}$$
$d_0$, $\mathbf{\Psi}_0$, $\bar{\mathbf{B}}_0$, and $\mathbf{\Omega}^{-1}_0$ are prior hyperparameters that you set as the researcher.
