I am trying to prove an inequality that : $$ \operatorname{det} \Sigma_{V \mathrm{~V}}<\prod i \operatorname{det}\left(\Sigma_{V_{i} V_{i}}\right) $$ Where $$ \Sigma_{V V}=\left[\begin{array}{ccc} \Sigma_{V_{1} V_{1}} & \cdots & \Sigma_{V_{1} V_{k}} \\ \vdots & \ddots & \vdots \\ \Sigma_{V_{k} V_{1}} & \cdots & \Sigma_{V_{k} V_{k}} \end{array}\right]$$ is a corvariance matrix, all of its block matrix is postive definite and Symmetric.
Any one may help me please? Thank you very much.
