 
  
  
  
  
 
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,
). For example,

Most users need not be aware
of the details, because ScaLAPACK routines are provided to work with this
representation:
 without forming Q explicitly.
 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
 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.
);  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
; hence  need not
be stored. In real arithmetic,
 need not
be stored. In real arithmetic,  , except that
, except that
 implies H = I.
 implies H = I.
In complex arithmetic ,  may be 
complex and satisfies
 may be 
complex and satisfies
 and
 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
.
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
 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 .
. 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].
 
  
  
  
 