So, given - $\vec{x}, \vec{u}$ and $\vec{v}$ which are N-dimensional, $\left<x_i\right> = 0$ for every $i$, and $\vec{u}$ is constant, there exist $y_1, y_2$ so that $y_1 = \vec{x}\vec{u}$ and $y_2=\vec{x}\vec{v}$. Also let us define $C=\left<\vec{x}\vec{x}^T\right>$.
I need to find $\vec{v}$ which minimizes $cov(y_1, y_2)$ and express it using $C$ and $\vec{u}$.
I will try to describe my try to solve this problem:
It came out that $cov(y_1, y_2)=\vec{u}^TC\vec{v}$.
I defined a Lagrange function as follows - $L(\vec{v}, \lambda) = \vec{u}^TC\vec{v} - \lambda(\vec{v}^T\vec{v}-1)$
So far so good, now -
$\frac{\partial L(\vec{v},\lambda)}{\partial \vec{v}}=\vec{u}^TC-2\lambda\vec{v}=0$
$\frac{\partial L(\vec{v},\lambda)}{\partial \lambda}=\vec{v}^T\vec{v}-1=0$
So that ${\vec{v}}=\frac{\vec{u}^TC}{2\lambda}$ and $\vec{v}^T\vec{v}=1$
I really couldn't find the correct move from this step. I have tried to extract $\lambda$ so I could express $\vec{v}$ without using $\lambda$ but I always get to a point I get true or null statements.
Any leads or ideas will be appreciated :D
Thanks!