I have a vector $X$ of random variables with mean $\mu$ and covariance matrix $\Sigma$.
I want to create a vector $Y$ of random variables that are uncorrelated, such that the original variables in $X$ are linear combinations of the variables in $Y$.
I remember vaguely that this requires diagonalizing $\Sigma$, but I am not sure about the details.
For concreteness, assume that:
$$ \Sigma = {1\over 36} \begin{bmatrix} 5 & -1 & -1 & -1 & -1 & -1 \\ -1 & 5 & -1 & -1 & -1 & -1 \\ -1 & -1 & 5 & -1 & -1 & -1 \\ -1 & -1 & -1 & 5 & -1 & -1 \\ -1 & -1 & -1 & -1 & 5 & -1 \\ -1 & -1 & -1 & -1 & -1 & 5 \end{bmatrix} $$
By diagonalizing $\Sigma$, I get matrices $C$ and $D$ such that $D$ is diagonal and $\Sigma = C D C^{-1}$:
$$ C = \left[\begin{matrix}1 & -1 & -1 & -1 & -1 & -1\\1 & 1 & 0 & 0 & 0 & 0\\1 & 0 & 1 & 0 & 0 & 0\\1 & 0 & 0 & 1 & 0 & 0\\1 & 0 & 0 & 0 & 1 & 0\\1 & 0 & 0 & 0 & 0 & 1\end{matrix}\right] \\ D= \left[\begin{matrix}0 & 0 & 0 & 0 & 0 & 0\\0 & \frac{1}{6} & 0 & 0 & 0 & 0\\0 & 0 & \frac{1}{6} & 0 & 0 & 0\\0 & 0 & 0 & \frac{1}{6} & 0 & 0\\0 & 0 & 0 & 0 & \frac{1}{6} & 0\\0 & 0 & 0 & 0 & 0 & \frac{1}{6}\end{matrix}\right] $$ How do I proceed from here - how do I express the vector $X$ as a function of a vector $Y$ where all variables in $Y$ are uncorrelated?
Is there maybe a different way to get uncorrelated variables from correlated ones?