 Solution

The Schur complement can be writtenbecause the first term is a conditional variance and ${\sigma }^2>0$. Therefore $c_1=s^{-1}>0$ and the squared term is nonnegative, soEquality holds exactly when the remaining conditional covariance $k(x_n,x_{\ast })-b^T{A}^{-1}a$ is zero. In particular, the reduction need not be strict even at a previously unseen input.