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Say I want a QR decomposition of matrix $A$, where orthogonality of $Q$ is with respect to a generic non-degenerate positive-definite bilinear form $\phi$ (in my case, $\phi$ is "defined" by a finite-element mass matrix). This seems like a standard thing one might want to do, so is there a Python implementation out there somewhere (that say, uses Modified Gram-Schmidt with pivoting, or uses Householder Reflections)? My matrix $A$ is tall, thin, and dense.

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Since the bilinear form is positive definite, you can use a Cholesky decomposition to write $M = W^T W$, where $\phi(x,y) = x^T M y$. Then compute the QR decomposition of a $W$-weighted $A$, i.e., $U R = W A$, and define $Q = W^{-1}U$. You obtain, $Q^T M Q = U^T W^{-T}W^T W W^{-1} U = I$, and $Q R = W^{-1}U R = A$.

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