I've got trouble with matrix optimization problem..
Proving the Principal Component Analysis (PCA) algorithm, we want to maximize the objective function $u^TSu$ subject to $u^Tu=1$, where $u$ is a principal component vector, $S$ is a scatter matrix (or covariance matrix).
In my textbook, they used the technique of Lagrange multipliers and just differentiated the equation with respect to u and set the result to zero.
Is it always guaranteed for that objective function to get maximized at extrema? If I differentiate $u^TSu$ twice, only $S$ remains, which is a semi-positive definite matrix. That result reminds me of a convex function, which is minimized at extrema.
Does anyone have any insight or experience with such a problem?