Uncovering Surprising Properties of the Cholesky Decomposition

[amateur82]

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?

 

[amateur82]

A diagonal matrix is trivially a fixed point of this process. (As for a diagonal matrix, A=B, and consequently, X=Y.) Numerical experiments show that all these sequences seem to converge to a diagonal matrix, yet the rate of convergence is different, even for matrices of same dimension. So perhaps this could be used as a 'grade' to rank matrices. For 2x2 matrices, the map is easy to give explicitly. If we denote an arbitrary member of sequence as X_(t), we have

x_(t+1)11 = x_(t)11 + x_(t)12² / x_(t)11
x_(t+1)12 = (x_(t)12 / √x_(t)11)√(x_(t)22 - x_(t)12² / x_(t)11)
x_(t+1)22 = x_(t)11 - x_(t)12² / x_(t)11.

So the map contracts x_(t+1)22 and increases x_(t+1)11 uniformly, and maintains the sign of x_(t+1)12. This wasn't true for off-diagonals of larger matrices, at least according to numerical trials. It would probably be easy to derive other properties for the Cholesky seqeunece in M(2x2); please feel free to try. I'm not expecting any major scientific breakthrough from here!