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:
\sigma_{1}^2 & \sigma_{12} \\ 
\sigma_{21} & \sigma_{2}^2

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

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 >