HP MLIB User's Guide Vol. 1 7th Ed.

38 HP MLIB User’s Guide
What you need to know to use vector subprograms
Representation of a Householder matrix
This section explains how the BLAS Standard represents a Householder
matrix.
An elementary reflector (or elementary Householder matrix) H of order n is a
unitary matrix of the form
where τ is a scalar, and υ is an n-vector, with
υ is often referred to as the Householder vector. Often, υ can have leading or
trailing zero elements, but for the purposes of this discussion, assume that H
has no special structure.
This representation sets υ1 = 1, meaning that υ1 need not be stored. In real
arithmetic, except that τ = 0 implies H = I. This representation agrees
with that used in LAPACK.
In complex arithmetic, τ may be complex, and satisfies
Thus a complex H is not Hermitian, as with other representations, but it is
unitary. The latter is the important property. The advantage of allowing τ to be
complex is that, given an arbitrary complex vector x, H can be computed so that
with real β. This is useful, for example, when reducing a complex Hermitian
matrix to a real symmetric tridiagonal matrix, or a complex rectangular matrix
to real bidiagonal form.
HIτυυ
H
–=
τ
2
υ
2
2
2Re τ()=
1 τ 2,≤≤
1 Re τ() 2 and τ 1–1≤≤≤
H
∗
x β 10… 0,, ,()
T
=