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<BR> <P>
<H1><A NAME="SECTION04340000000000000000">Orthogonal or Unitary Matrices</A></H1>
<A NAME="secorthog"> </A>
<P>
A real orthogonal or complex unitary matrix (usually denoted <I>Q</I>) is often
represented<A NAME="2189"> </A> in
ScaLAPACK as a product of <B>elementary reflectors</B> -- also referred to as
<A NAME="2191"> </A>
<A NAME="2192"> </A>
<B>elementary Householder matrices</B> (usually denoted <IMG WIDTH=18 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline14421" SRC="img231.gif">). For example,
<BR><IMG WIDTH=316 HEIGHT=16 ALIGN=BOTTOM ALT="displaymath14415" SRC="img232.gif"><BR>
Most users need not be aware
of the details, because ScaLAPACK routines are provided to work with this
representation:
<P>
<UL>
<LI> routines whose names begin PSORG- (real) or PCUNG- (complex) can generate
all or part of <I>Q</I> explicitly;
<LI> routines whose name begin PSORM- (real) or PCUNM- (complex) can multiply
a given matrix by <I>Q</I> or <IMG WIDTH=25 HEIGHT=31 ALIGN=MIDDLE ALT="tex2html_wrap_inline13302" SRC="img124.gif"> without forming <I>Q</I> explicitly.
<P>
</UL>
<P>
The following details may occasionally be useful.
<P>
An elementary reflector (or elementary Householder matrix) <I>H</I> of order
<I>n</I> is a
unitary matrix<A NAME="2201"> </A> of the form
<A NAME="2202"> </A>
<A NAME="2203"> </A>
<BR><A NAME="Hdef"> </A><IMG WIDTH=500 HEIGHT=20 ALIGN=BOTTOM ALT="equation2204" SRC="img233.gif"><BR>
where <IMG WIDTH=9 HEIGHT=8 ALIGN=BOTTOM ALT="tex2html_wrap_inline14435" SRC="img234.gif"> is a scalar and <I>v</I> is an <I>n</I>-vector, with
<IMG WIDTH=129 HEIGHT=31 ALIGN=MIDDLE ALT="tex2html_wrap_inline14441" SRC="img235.gif">); <I>v</I> is often referred to
as the <B>Householder vector.</B><A NAME="2210"> </A>
Often <I>v</I> has several leading or trailing zero elements, but for the
purpose of this discussion assume that <I>H</I> has no such special structure.
<P>
Some redundancy in the representation (<A HREF="node66.html#Hdef">3.4</A>) 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 <IMG WIDTH=46 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline14449" SRC="img236.gif">; hence <IMG WIDTH=14 HEIGHT=17 ALIGN=MIDDLE ALT="tex2html_wrap_inline14451" SRC="img237.gif"> need not
be stored. In real arithmetic, <IMG WIDTH=72 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline14453" SRC="img238.gif">, except that
<IMG WIDTH=41 HEIGHT=13 ALIGN=BOTTOM ALT="tex2html_wrap_inline14455" SRC="img239.gif"> implies <I>H</I> = <I>I</I>.
<P>
In complex arithmetic<A NAME="2212"> </A>, <IMG WIDTH=9 HEIGHT=8 ALIGN=BOTTOM ALT="tex2html_wrap_inline14435" SRC="img234.gif"> may be
complex and satisfies
<IMG WIDTH=106 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline14461" SRC="img240.gif"> and <IMG WIDTH=77 HEIGHT=27 ALIGN=MIDDLE ALT="tex2html_wrap_inline14463" SRC="img241.gif">.
Thus a complex <I>H</I> is
not Hermitian (as it is in other representations), but it is unitary,
which is the important property. The advantage of allowing <IMG WIDTH=9 HEIGHT=8 ALIGN=BOTTOM ALT="tex2html_wrap_inline14435" SRC="img234.gif"> to be
complex is that, given an arbitrary complex vector <I>x</I>, <I>H</I> can be computed
so that <BR><IMG WIDTH=334 HEIGHT=21 ALIGN=BOTTOM ALT="displaymath14416" SRC="img242.gif"><BR>
with <I>real</I> <IMG WIDTH=11 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline14473" SRC="img243.gif">. This is useful, for example,
when reducing a complex Hermitian matrix to real symmetric tridiagonal form<A NAME="2215"> </A>
or a complex rectangular matrix to real bidiagonal form<A NAME="2216"> </A>.
<P>
For further details, see Lehoucq [<A HREF="node189.html#lawn72">94</A>].
<P>
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<P><ADDRESS>
<I>Susan Blackford <BR>
Tue May 13 09:21:01 EDT 1997</I>
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