Cholesky LDLT decomposition and B-orthogonalization

Question

I know that it is possible to B-orthogonalize a matrix X via a standard LL^T decomposition:

1) M = X^TBX

2) U = chol(M)

3) answer = XU^-1

How can I use LDL^T decomposition for this?

Answer 1

For a dense ${X}_{n \times k}$ and a sparse symmetric positive-definite matrix ${B}_{n \times n}$, we wish to compute a dense matrix S of the same dimensions and column space as X such that: S^TBS = Identity // this is a defenition of the B-orthogonalisation

And we have the LDLT-decomposition only. I will use U instead of L^T

Then, X^TBX = U^TDU = U^TD^1/2D^1/2U = U^TD^1/2S^TBSD^1/2U

Therefore, X = SD^1/2U ===> S = X(D^1/2U)^-1 = XU^-1(D^-1)^1/2