I noticed a funny thing. The Cholesky decomposition can be defined as X=AB, where A is lower triangular. Generally Y=BA is not X, but Y seems to be a positive definite matrix. I wonder if there is any special properties to the pair (X,Y). I know that a positive definite matrix can be interpreted as a metric. So a pair of conjugate metrics? It is also funny that the product AB is not commutative. You would think so, since A=B'. So when you map by X, first you turn to direction B, and then to orthogonal direction. For some reason this seems to be completely different than turning first to orthogonal direction and then to direction B... edit: played around more, and found out, that it's not actually a pair, but a sequence of positive definite matrices! chol(Y) doesn't involve A and B, but some other triangular matrices. So a map from psd matrix to another X -> Y -> Z... does anybody know where this sequence leads?