Convexity of a Certain Set of Covariance Matrices - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T11:34:56Z http://mathoverflow.net/feeds/question/119348 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/119348/convexity-of-a-certain-set-of-covariance-matrices Convexity of a Certain Set of Covariance Matrices Ziv Goldfeld 2013-01-19T17:59:48Z 2013-01-19T17:59:48Z <p>Hello,</p> <p>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: $\Sigma_{12}=\Sigma_{1U}\Sigma_U^{-1}\Sigma_{U2}$.</p> <p>Let us define the matrix:</p> <p>$\Sigma=\left( \begin{array}{ccc} \Sigma_1 &amp; \Sigma_{1U} &amp; A\\ \Sigma_{1U}^T &amp; \Sigma_U &amp; \Sigma_{2U}^T\\ A^T &amp; \Sigma_{2U} &amp; \Sigma_{2}\end{array}\right)$.</p> <p>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?</p> <p>Thank you all in advance,</p> <p>Best regards!</p>