Significant Properties of Householder Transformation

Abstract

A Householder transformation is a linear operator, which reflects any vector with respect to a given plane. This review article presents an idea behind the definition of this transformation and discusses the significant properties of the transformation from an operator-theory point of view. Algebraic properties of this transformation are its being unitary, Hermitian and involuntary. Its geometric properties include norm-orthogonality-angle preservings. Moreover, we show that every unitary matrix can be written as a product of elementary reflectors, which are in the form related to the Householder one.