In [1, p. 60 exercise 3.1] a $2 \times 1$ real random vector $\mathbf{x}$ is given , which is distributed according to a multivatiate Gaussian PDF with zero mean and covariance matrix:
 $\mathbf{C}_{xx}= \left[ \begin{array}{cc} \sigma_{1}^2 & \sigma_{12} \\ \sigma_{21} & \sigma_{2}^2 \end{array}\right]$

We are asked to find $\alpha$ if $\mathbf{y}=\mathbf{L}^{-1}\mathbf{x}$ where $\mathbf{L}^{-1}$ is given by the relation:

 $y_{1}$ $=$ $x_{1}$ $y_{2}$ $=$ $\alpha x_{1}+x_{2}$

so that $y_{1}$ and $y_{2}$ are uncorrelated and hence independent. We are also asked to find the Cholesky decomposition of $\mathbf{C}_{xx}$ which expresses $\mathbf{C}_{xx}$ as $\mathbf{L}\mathbf{D}\mathbf{L}^{T}$, where $\mathbf{L}$ is lower triangular with 1′s on the principal diagonal and $\mathbf{D}$ is a diagonal matrix with positive diagonal elements. read the conclusion >