Orthogonalise a matrix with respect to another matrix

Question

I need to do a particular operation and I am not able to find a name or some ideas to make the operation happen.

I have a matrix $F_s \in \mathbb{R}^{n \times n_s}$. I do an Singular value decomposition on it to get the left singular vectors $F_s \in \mathbb{R}^{n \times m}$.

Here $F_u^T * F_u = I$. I want to now Orthogonalise this $F_u$ with respect to $Q \in \mathbb{R}^{n \times n}$ i.e., $F_u^T Q F_u = I$.

A very similar question has been asked before here, but no proper response is obtained.

I am looking for the process I need to do to orthogonalize a matrix $F_u$ with respect to another matrix $Q$. I looked at Generalised SVD and Weighted SVD, but they dont fit with this idea.

Thank You.

Answer 1

Presumably, $Q$ is some kind of positive definite matrix.

Perform a Cholesky decomposition to get $Q = U^TU$. We then have $$ F_u^TQF_u = I \iff (UF_u)^T(UF_u) = I $$ That is, your process is equivalent to "orthogonalising" $UF_u$.