An elementary reflector (also called Householder transformation) is a unitary matrix of the form \(H = I-2vv^\star\), where \(I\) is the identity matrix and \(v\) is some vector with \(\| v\| _2=1\). For any two vectors \(p\) and \(q\) with \(\| p\| _2 = \| q\| _2\) there exists such a transformation which maps \(p\) to \(q\). This mapping property makes elementary reflectors an important tool for developing algorithms in numerical linear algebra.
Structure-preserving algorithms often rely on variants of elementary reflectors which yield, e.g., complex orthogonal, symplectic or pseudo-orthogonal transformations. This paper provides a general and comprehensive framework for such variants. If \(\langle \cdot, \cdot\rangle\) is a bilinear form (\(\langle \cdot, \cdot\rangle = x^T My\) for some matrix \(M\)) or a sesquilinear form (\(\langle \cdot, \cdot\rangle = x^\star My\) for some matrix \(M\)) then the automorphism group \(\mathbb G\) associated with \(\langle \cdot, \cdot\rangle\) consists of all matrices \(G\) satisfying \(\langle G x, Gy\rangle = \langle x, y\rangle\). A \(\mathbb G\)-reflector is defined as a transformation \(G = I + uv^\star\) with \(G \in \mathbb G\). It is shown that such a \(G\) can always be written in the form \(G = I + \beta uu^T\) (bilinear \(\langle \cdot, \cdot\rangle\)) or \(G = I + \beta uu^\star\) (sesquilinear \(\langle \cdot, \cdot\rangle\)), provided that \(M\) is nonsingular. The usefulness of \(\mathbb G\)-reflectors for dealing with structured matrices stems from the fact that the transformation \(A\rightarrow G^{-1} A G\) not only preserves the automorphism group but also the Jordan and Lie algebras associated with \(\langle \cdot, \cdot\rangle\). For example, if \(\langle x,y \rangle = x^\star y\) then \(\mathbb G\) consists of unitary matrices and the transformation above preserves Hermitian as well as skew-Hermitian matrices.
This paper gives geometric interpretations of \(G\) depending on the question whether \(u\) is isotropic (\(\langle u, u \rangle = 0\)). Isotropic vectors also limit the mapping capabilities of \(G\); a non-isotropic vector can never be mapped to an isotropic vector; and vice versa. On the other hand, for (skew-)symmetric bilinear and sesquilinear forms, this paper shows that there always exists a unique \(\mathbb G\)-reflector \(G\) that maps \(p\) to \(q\) if \(\langle p,p \rangle = \langle q,q \rangle\) and \(\langle q-p,p\rangle \not= 0\). Various formulas for computing and representing \(G\) are given. These formulas also provide insight into the construction of \(G\) for the standard case \(\langle x,y \rangle = x^\star y\).
This paper is a very useful guide for developing and understanding structure-preserving algorithms.