Suppose that I have an $(n\times 1)$ vector of random variables, $\varepsilon$. However, I know that $k$ linear combinations of $\varepsilon$ are 0. Specifically, I know that for a $(k\times n)$ matrix $B$, $$ B\varepsilon = 0 $$ I want to learn about the restrictions this imposes on the covariance matrix of $\varepsilon$, $\Sigma$. Suppose I further know the variances of each $\varepsilon$, i.e. I know the diagonal of $\Sigma$. I also impose the normalization that the first $(n-k)$ shocks are uncorrelated with each other. Intuitively, it would be that the remaining $k$ shocks are linear combinations of the first $(n-k)$ shocks. I want to find a Cholesky factorization $LL^{T}$ of $\Sigma$ that satisfies these restrictions but I do not know where to start.
Essentially, I want to find $L$ such that $LL^{T}=\Sigma$ and $BLL^{T}B^{T}=0$ imposing the above normalizations.
