Hello,

My question is about a certain set of matrices being convex or not. I'll start with some preliminaries in order to define myself properly. Let $X_1,U,X_2$ be three zero-mean Gaussian random vectors (RVs) of dimension $N\times 1$, that admit the Markov relation $X_1-U-X_2$. Let us use the notations: $\Sigma_i=\mathbb{E}[X_iX_i^T]$, for $i=1,2$, $\Sigma_U=\mathbb{E}[UU^T]$, $\Sigma_{iU}=\mathbb{E}[X_iU^T]$, for $i=1,2$ and finally $\Sigma_{12}=\mathbb{E}[X_1X_2^T]$. The Markov relation is equivalent to the fact that the auto- and cross- covariance matrices of the RV satisfy: $\Sigma_{12}=\Sigma_{1U}\Sigma_U^{-1}\Sigma_{U2}$.

Let us define the matrix:

$\Sigma=\left( \begin{array}{ccc} \Sigma_1 & \Sigma_{1U} & A\\\ \Sigma_{1U}^T & \Sigma_U & \Sigma_{2U}^T\\\ A^T & \Sigma_{2U} & \Sigma_{2}\end{array}\right) $.

where $A=\Sigma_{1U}\Sigma_{U}^{-1}\Sigma_{U2}$. My question regards the set of all legitimate matrices $\Sigma$, is this set convex? How can one check this?

Thank you all in advance,

Best regards!