# Decomposition of a random vector into uncorrelated components

I have a set of random vectors $Y_i$ and their correlation matrix $C_{i,j}$. Each vector can be thought of as a sum of two uncorrelated vectors $Y_i=A_iX+B_iY$, where $X,Y$ are the same vectors for each $Y_i$ and $A_i,B_i$ are some scalar coefficients. One can assume that $X$ and $Y$ are uncorrelated. The goal is to find all $A_i$, $B_i$ (for every $i$).

So far I was thinking to use the cholesky decomposition $D$ of the correlation matrix $C$ so that $(A_iX+B_iY)D=C$, and then solve the system for A and B. The problem is, however, that I do have two more variables than equations. So, the question is: how to use the fact that X and Y are uncorrelated to exclude them from the picture and just find A and B? Or is there other method to solve the problem? Thanks to everyone!