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<H1><A NAME="SECTION03540000000000000000"></A><A NAME="secorthog"></A>
<BR>
Representation of Orthogonal or Unitary Matrices
</H1>
<P>
A real orthogonal or complex unitary matrix (usually denoted <B><I>Q</I></B>) is often
represented<A NAME="19987"></A> in
LAPACK as a product of <B>elementary reflectors</B> -- also referred to as
<A NAME="19989"></A>
<A NAME="19990"></A>
<B>elementary Householder matrices</B> (usually denoted <B><I>H</I><SUB><I>i</I></SUB></B>). For example,
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
Q = H_{1} H_{2} \ldots H_{k}.
\end{displaymath}
-->
<IMG
WIDTH="133" HEIGHT="30" BORDER="0"
SRC="img938.png"
ALT="\begin{displaymath}Q = H_{1} H_{2} \ldots H_{k}. \end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
Most users need not be aware
of the details, because LAPACK routines are provided to work with this
representation:
<P>
<UL><LI>routines whose names begin SORG- (real) or CUNG- (complex) can generate
all or part of <B><I>Q</I></B> explicitly;
<P>
<LI>routines whose name begin SORM- (real) or CUNM- (complex) can multiply
a given matrix by <B><I>Q</I></B> or <B><I>Q</I><SUP><I>H</I></SUP></B> without forming <B><I>Q</I></B> explicitly.
<P>
</UL>
<P>
The following further details may occasionally be useful.
<P>
An elementary reflector (or elementary Householder matrix) <B><I>H</I></B> of order
<B><I>n</I></B> is a
unitary matrix<A NAME="19999"></A> of the form
<A NAME="20000"></A>
<A NAME="20001"></A>
<BR>
<DIV ALIGN="RIGHT">
<!-- MATH
\begin{equation}
H = I - \tau v v^{H}
\end{equation}
-->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="Hdef"></A><IMG
WIDTH="109" HEIGHT="28" BORDER="0"
SRC="img939.png"
ALT="\begin{displaymath}
H = I - \tau v v^{H}
\end{displaymath}"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(5.1)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
where <IMG
WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img220.png"
ALT="$\tau$">
is a scalar, and <B><I>v</I></B> is an <B><I>n</I></B>-vector, with
<!-- MATH
$| \tau | ^2 || v || _2 ^2 = 2 \rm {Re}(\tau$
-->
<IMG
WIDTH="136" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
SRC="img940.png"
ALT="$\vert \tau \vert ^2 \vert\vert v \vert\vert _2 ^2 = 2 \rm {Re}(\tau$">); <B><I>v</I></B> is often referred to
as the <B>Householder vector</B><A NAME="20008"></A> .
Often <B><I>v</I></B> has several leading or trailing zero elements, but for the
purpose of this discussion assume that <B><I>H</I></B> has no such special structure.
<P>
There is some redundancy in the representation (<A HREF="node128.html#Hdef">5.1</A>), which can be
removed in
various ways. The representation used in LAPACK (which differs from
those used in LINPACK or EISPACK) sets <B><I>v</I><SUB>1</SUB> = 1</B>; hence <B><I>v</I><SUB>1</SUB></B> need not
be stored. In real arithmetic,
<!-- MATH
$1 \leq \tau \leq 2$
-->
<IMG
WIDTH="78" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img941.png"
ALT="$1 \leq \tau \leq 2$">,
except that
<IMG
WIDTH="46" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img942.png"
ALT="$\tau = 0$">
implies <B><I>H</I> = <I>I</I></B>.
<P>
In complex arithmetic<A NAME="20010"></A>, <IMG
WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img220.png"
ALT="$\tau$">
may be
complex, and satisfies
<!-- MATH
$1 \leq \rm {Re}(\tau) \leq 2$
-->
<IMG
WIDTH="112" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img943.png"
ALT="$1 \leq \rm {Re}(\tau) \leq 2$">
and
<!-- MATH
$| \tau - 1 | \leq 1$
-->
<IMG
WIDTH="86" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img944.png"
ALT="$\vert \tau - 1 \vert \leq 1$">.
Thus a complex <B><I>H</I></B> is
not Hermitian (as it is in other representations), but it is unitary,
which is the important property. The advantage of allowing <IMG
WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img220.png"
ALT="$\tau$">
to be
complex is that, given an arbitrary complex vector <B><I>x</I></B>, <B><I>H</I></B> can be computed
so that <BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
H^H x = \beta (1, 0, \ldots , 0)^T
\end{displaymath}
-->
<IMG
WIDTH="168" HEIGHT="31" BORDER="0"
SRC="img945.png"
ALT="\begin{displaymath}H^H x = \beta (1, 0, \ldots , 0)^T \end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
with <I>real</I> <IMG
WIDTH="15" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img50.png"
ALT="$\beta$">.
This is useful, for example,
when reducing a complex Hermitian matrix to real symmetric tridiagonal form<A NAME="20013"></A>,
or a complex rectangular matrix to real bidiagonal form<A NAME="20014"></A>.
<P>
For further details, see Lehoucq [<A
HREF="node151.html#lawn72">79</A>].
<P>
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<ADDRESS>
<I>Susan Blackford</I>
<BR><I>1999-10-01</I>
</ADDRESS>
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