I understand that when one have multiple independent variable that follows $N\sim(0,1)$, denoted as $A$ if we have a correlation matrix $R$, we can generate correlated variables $B$ that are normally distributed using the Cholesky decomposition as stated here: Given the Cholesky decomposition of $R$ be $$ R=LL^t $$ Then we have $$ LA=B $$
However, is there any way for one to simulate correlated $\chi^2$ given multiple independent $\chi^2(1)$ variables?
I have tried to apply the cholesky directly to the $R^2$ and solve the following: $$ R^2=LL^t $$ $$ LA^2=B^2 $$ which doesn't give the expected result because $B^2$ isn't $\chi^2$ distributed although they do have the expected correlation $R^2$. I have also try to squaring the whole equation such that I have $$ LAA^tL^t =BB^t $$ Yet the result is difficult to interpret as $BB^t$ is a square matrix.
So is there any way to generate correlated $\chi^2$ variables given multiple independent $\chi^2(1)$ variables? Or will $A$ need to be following some other distribution for $B$ to be correlated $\chi^2$ distribution?