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<H3><A NAME="SECTION034111400000000000000"></A><A NAME="sec_singular"></A>
<BR>
Singular Eigenproblems
</H3>
<P>
In this section, we give a brief discussion of singular
matrix pairs <B>(<I>A</I>,<I>B</I>)</B><A NAME="12856"></A>.
<P>
If the determinant of <IMG
WIDTH="63" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img56.png"
ALT="$A - \lambda B$">
is zero for all values of <IMG
WIDTH="15" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img23.png"
ALT="$\lambda$">
(or the determinant of
<!-- MATH
$\beta A - \alpha B$
-->
<IMG
WIDTH="75" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img851.png"
ALT="$\beta A - \alpha B$">
is zero for all
<!-- MATH
$(\alpha,\beta)$
-->
<IMG
WIDTH="48" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img181.png"
ALT="$(\alpha,\beta)$">),
the pair <B>(<I>A</I>,<I>B</I>)</B> is said to be <B>singular</B>.
The eigenvalue problem of a singular pair is much more complicated
than for a regular pair.
<P>
Consider for example the singular pair
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
A = \left( \begin{array}{cc}
1 & 0 \\0 & 0 \\
\end{array} \right), \quad\quad
B = \left( \begin{array}{cc}
1 & 0 \\0 & 0 \\
\end{array} \right),
\end{displaymath}
-->
<IMG
WIDTH="275" HEIGHT="54" BORDER="0"
SRC="img852.png"
ALT="\begin{displaymath}
A = \left( \begin{array}{cc}
1 & 0 \\ 0 & 0 \\
\end{array...
...t( \begin{array}{cc}
1 & 0 \\ 0 & 0 \\
\end{array} \right),
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
which has one finite eigenvalue 1 and one indeterminate eigenvalue 0/0.
To see that neither eigenvalue is well determined by the data,
consider the slightly different problem
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
A' = \left( \begin{array}{cc}
1 & \epsilon_1 \\\epsilon_2 & 0 \\
\end{array} \right), \quad\quad
B' = \left( \begin{array}{cc}
1 & \epsilon_3 \\\epsilon_4 & 0 \\
\end{array} \right),
\end{displaymath}
-->
<IMG
WIDTH="308" HEIGHT="54" BORDER="0"
SRC="img853.png"
ALT="\begin{displaymath}
A' = \left( \begin{array}{cc}
1 & \epsilon_1 \\ \epsilon_2 ...
...c}
1 & \epsilon_3 \\ \epsilon_4 & 0 \\
\end{array} \right),
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
where the <IMG
WIDTH="17" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img854.png"
ALT="$\epsilon_i$">
are tiny nonzero numbers. Then it is easy to see
that <B>(<I>A</I>',<I>B</I>')</B> is regular with eigenvalues
<!-- MATH
$\epsilon_1 / \epsilon_3$
-->
<IMG
WIDTH="42" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img855.png"
ALT="$\epsilon_1 / \epsilon_3$">
and
<!-- MATH
$\epsilon_2 / \epsilon_4$
-->
<IMG
WIDTH="42" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img856.png"
ALT="$\epsilon_2 / \epsilon_4$">.
Given <EM>any</EM> two complex numbers <IMG
WIDTH="22" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img857.png"
ALT="$\lambda_1$">
and
<IMG
WIDTH="22" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img858.png"
ALT="$\lambda_2$">,
we can find arbitrarily tiny <IMG
WIDTH="17" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img854.png"
ALT="$\epsilon_i$">
such that
<!-- MATH
$\lambda_1 = \epsilon_1 / \epsilon_3$
-->
<IMG
WIDTH="83" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img859.png"
ALT="$\lambda_1 = \epsilon_1 / \epsilon_3$">
and
<!-- MATH
$\lambda_2 = \epsilon_2 / \epsilon_4$
-->
<IMG
WIDTH="83" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img860.png"
ALT="$\lambda_2 = \epsilon_2 / \epsilon_4$">
are the eigenvalues of <B>(<I>A</I>',<I>B</I>')</B>.
Since, in principle, roundoff could change <B>(<I>A</I>,<I>B</I>)</B> to <B>(<I>A</I>',<I>B</I>')</B>, we
cannot hope to compute accurate or even meaningful eigenvalues of
singular problems, without further information.
<P>
<P>
It is possible for a pair <B>(<I>A</I>,<I>B</I>)</B> in Schur form to be very close to singular,
and so have very sensitive eigenvalues, even if no diagonal entries of
<B><I>A</I></B> or <B><I>B</I></B> are small.
It suffices, for example, for <B><I>A</I></B> and <B><I>B</I></B> to nearly have a common null space
(though this condition is not necessary).
For example, consider the 16-by-16 matrices
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
A'' = \left( \begin{array}{ccccc}
0.1 & 1 & & & \\
& 0.1 & 1 & & \\
& & \ddots & \ddots & \\
& & & 0.1 & 1 \\
& & & & 0.1 \\
\end{array} \right) \quad\quad
\mbox{and} \quad\quad B'' = A''.
\end{displaymath}
-->
<IMG
WIDTH="432" HEIGHT="124" BORDER="0"
SRC="img861.png"
ALT="\begin{displaymath}
A'' = \left( \begin{array}{ccccc}
0.1 & 1 & & & \\
& 0.1 ...
...end{array} \right) \quad\quad
\mbox{and} \quad\quad B'' = A''.
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
Changing the <B>(<I>n</I>,1)</B> entries of <B><I>A</I>''</B> and <B><I>B</I>''</B> to <B>10<SUP>-16</SUP></B>, respectively,
makes both <B><I>A</I></B> and <B><I>B</I></B> singular, with a common null vector.
Then, using a technique analogous to the one applied to <B>(<I>A</I>,<I>B</I>)</B> above,
we can show that there is a
perturbation of <B><I>A</I>''</B> and <B><I>B</I>''</B> of norm
<!-- MATH
$10^{-16} + \epsilon$
-->
<IMG
WIDTH="75" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
SRC="img862.png"
ALT="$10^{-16} + \epsilon$">,
for <EM>any</EM> <IMG
WIDTH="44" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img863.png"
ALT="$\epsilon>0$">,
that makes the 16 perturbed eigenvalues
have <EM>any</EM> arbitrary 16 complex values.
<P>
A complete understanding of the structure of a singular eigenproblem
<B>(<I>A</I>,<I>B</I>)</B> requires a study of its <EM>Kronecker canonical form</EM>,
a generalization of the <EM>Jordan canonical form</EM>.
In addition to Jordan blocks for finite and infinite eigenvalues,
the Kronecker form can contain ``singular blocks'', which occur only
if
<!-- MATH
${\rm det}(A- \lambda B) \equiv 0$
-->
<IMG
WIDTH="133" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img864.png"
ALT="${\rm det}(A- \lambda B) \equiv 0$">
for all <IMG
WIDTH="15" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img23.png"
ALT="$\lambda$">
(or if <B><I>A</I></B> and <B><I>B</I></B>
are nonsquare).
See [<A
HREF="node151.html#gantmacher">53</A>,<A
HREF="node151.html#stewart72">93</A>,<A
HREF="node151.html#wilkinson79">105</A>,<A
HREF="node151.html#vandooren79">97</A>,<A
HREF="node151.html#demmelkagstrom87">29</A>] for
more details. Other numerical software, called GUPTRI<A NAME="12891"></A>,
is available for
computing a generalization of the
Schur canonical form for singular eigenproblems
[<A
HREF="node151.html#demmelkagstrom93a">30</A>,<A
HREF="node151.html#demmelkagstrom93b">31</A>].
<P>
The error bounds discussed in this guide hold for regular pairs only
(they become unbounded, or otherwise provide no information, when
<B>(<I>A</I>,<I>B</I>)</B> is close to singular). If a (nearly) singular pencil is reported
by the software discussed in this guide, then a further study of the matrix
pencil should be conducted, in order to determine whether meaningful results
have been computed.
<P>
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<ADDRESS>
<I>Susan Blackford</I>
<BR><I>1999-10-01</I>
</ADDRESS>
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