Real tridiagonalization of Hermitian matrices by modified Householder transformation (Q1817248)

Elementary Householder reflections are of the form \(I-\sigma ww^H\), where \(\sigma\in\mathbb{R}\) is even when \(w\in\mathbb{C}^n\). This short note considers extended reflections where \(\sigma\in\mathbb{C}\). It is shown that one can always transform any complex vector to a real multiple of the first unit vector using an extended reflection. As a consequence, Hermitian matrices can be reduced to real, symmetric tridiagonal form via two-sided transformations with a series of extended reflections. A similar result is shown to hold if the vector components are quaternions.