let me explain you the process, I have random variables in a matrix $X_1$: $260\times3$.
I have my correlation matrix $\rho_1$: $3\times3$ from my matrix $X_1$.
Now I use a Cholesky decomposition and I create $X_2$ from the covariance in $X_1$.
As expected correlation $\rho_2$ of $X_2$ and $\rho_1$ of $X_1$ are very close.
When I create $X_3$ as the $X_1$ in which the first column has been replaced by the first one in $X_2$, the correlation matrix is very far from $\rho_1$ and $\rho_2$.
1) I don’t really get why as my correlation matrix are very close between $X_1$ and $X_2$ so I was expected $X_3$ to have a correlation close to the two other ?
2) Is there a way to perform this simulation / using Cholesky to in the end keep the two last column of $X_1$ and recreate the first one while keeping the correlation (or very close) of $X_1$ ?
Thank you for your help