  Householder orthogonalization plays an important role in numerical linear
algebra. It attains perfect orthogonality regardless of the conditioning of the
input. However, in the context of a non-standard inner product, it becomes
difficult to apply Householder orthogonalization due to the lack of an initial
orthogonal basis. We propose strategies to overcome this obstacle and discuss
algorithms and variants of Householder orthogonalization with a non-standard
inner product. Rounding error analysis and numerical experiments demonstrate
that our approach is numerically stable.
