Let us consider a dynamic system $x_t = A x_{t-1}+v_t$ where $v_t$ is multivariate normal noise with zero mean, i.e. $v_t\sim\mathcal{N}(0,\Sigma)$ and $A$ is a matrix. As far as I know, for some $A$, the system is not stable, namely when there is an eigenvalue larger than zero.
My question: how to sample a random matrix $A$ so the system is stable.
Here is a way to generate "random" $n \times n$ matrices with all singular values $< 1$. Write $A = U \Sigma V^T$ where $U$ and $V^T$ are orthogonal matrices and $\Sigma$ is diagonal. For the diagonal entries of $\Sigma$, take a sample of size $n$ from the uniform distribution on $(0,1)$. To generate random orthogonal matrices, you could use e.g. these methods.