Given a random vector c with zero mean, the covariance matrix \(\Sigma = E[cc^T]\). We give a one line proof that it is positive semidefinite.
\[\begin{align} u^T\Sigma u = u^TE[cc^T]u = E[u^Tcc^Tu] = E[\|c^Tu\|^2] \ge 0. \end{align}\]Given a random vector c with zero mean, the covariance matrix \(\Sigma = E[cc^T]\). We give a one line proof that it is positive semidefinite.
\[\begin{align} u^T\Sigma u = u^TE[cc^T]u = E[u^Tcc^Tu] = E[\|c^Tu\|^2] \ge 0. \end{align}\]