This question is related to my previous question in here
So I was trying to simulate correlated $\chi^2(1)$ random variables given the desired co-variance matrix. However, it seems like the only possible route was the following
Given the co-variance matrix $R$, then we perform the cholesky decomposition such that $$ R=LL^t $$ Then we simulate a number of independent vector $A$ with $N\sim(0,1)$, we can then simulate $B$ using $$ LA=B $$ where B should be correlated $N\sim(0,1)$, then we can obtain the correlated $\chi^2(1)$ variables.
When playing around with the variables, I thought that given $$ LA=B $$ and that the correlated $\chi^2$ are simply $B^2$, then by squaring the equation, I should be able to get the corresponding independent $\chi^2$ variables e.g. $$ LAA^tL^t=BB^t $$ However, I note that $AA^t$ and $BB^t$ are both square matrix, so I am not sure whether if they are both $\chi^2$ distributed. What interested me most is that: given a set of correlated random variables that are normally distributed, we can simply use the cholesky distribution to orthogonalizing the variables or un-correlate them. Will there be similar simple equations that we can use to perform the same trick in orthogonalizing $\chi^2$ or even non-centric $\chi^2$ variables?