Problem with Lagrange multiplier

Question

I am trying to find the minimum & maximum value of f. G is a square matrix. Applying Lagrange multiplier method,

How should I define the term '?', and how could I calculate the partial derivatives?

Answer 1

You want to minimize (and maximize) $$ \Vert G u \Vert^2 \ \text{ subject to } \ \Vert u \Vert^2=1. $$ The Lagrangien is given by $$ \mathcal L(u,\lambda) = \Vert G u \Vert^2 - \lambda( \Vert u \Vert^2 - 1), $$ and the first order condition is $$ \frac{\partial \mathcal L}{\partial u} (u, \lambda) = 2 G^\top Gu - 2 \lambda u $$ meaning that $\frac{\partial \mathcal L}{\partial u} (u, \lambda) = 0$ is possible for $u \neq 0$ only if $u$ is an eigenvector of $G^\top G$, in which case $\lambda$ is the corresponding eigenvalue.