A real orthogonal or complex unitary matrix (usually denoted *Q*) is often
represented in
ScaLAPACK as a product of **elementary reflectors** -- also referred to as
**elementary Householder matrices** (usually denoted ). For example,

Most users need not be aware
of the details, because ScaLAPACK routines are provided to work with this
representation:

- routines whose names begin PSORG- (real) or PCUNG- (complex) can generate
all or part of
*Q*explicitly; - routines whose name begin PSORM- (real) or PCUNM- (complex) can multiply
a given matrix by
*Q*or without forming*Q*explicitly.

The following details may occasionally be useful.

An elementary reflector (or elementary Householder matrix) *H* of order
*n* is a
unitary matrix of the form

where is a scalar and *v* is an *n*-vector, with
); *v* is often referred to
as the **Householder vector.**
Often *v* has several leading or trailing zero elements, but for the
purpose of this discussion assume that *H* has no such special structure.

Some redundancy in the representation (3.4) exists, which can be
removed in
various ways. Like LAPACK, the representation used in ScaLAPACK (which
differs from
that used in LINPACK or EISPACK) sets ; hence need not
be stored. In real arithmetic, , except that
implies *H* = *I*.

In complex arithmetic , may be
complex and satisfies
and .
Thus a complex *H* is
not Hermitian (as it is in other representations), but it is unitary,
which is the important property. The advantage of allowing to be
complex is that, given an arbitrary complex vector *x*, *H* can be computed
so that

with *real* . This is useful, for example,
when reducing a complex Hermitian matrix to real symmetric tridiagonal form
or a complex rectangular matrix to real bidiagonal form .

For further details, see Lehoucq [94].

Tue May 13 09:21:01 EDT 1997