I want to make a covariance matrix that incorporates my belief of how correlated the various dimensions are. The reason why I want to incorporate my belief is that in my modelling, the dimensions are food items and I want to manually correlate which food items are similar to which other in my covariance matrix. Is there any way to generate a valid covariance matrix(ensuring positive definiteness, though symmetry is easy)? After making such a covariance matrix, I intend to do a Cholesky decomposition and use it to uncorrelate my data-points. Are there any ways to do an approximate Cholesky decomposition, given a non-positive definite symmetric matrix?
Any help would be appreciated!!!