Choleski decomposition and MGARCH

[Econometrician]

Does anyone know how to orthogonalize two residual series, X and Y (which may be residuals generated from the estimation of a bivariate GARCH model) using a Choleski decomposition, under the assumption that one of the residual series, say Y is partly driven by a purely idiosyncratic shock and also by X? Am a Masters student and my thesis depends on this, so I will really appreciate any help possible.