On Thu, Apr 16, 2009 at 3:05 PM, Chetan Jhurani <span dir="ltr"><<a href="mailto:chetan@ices.utexas.edu">chetan@ices.utexas.edu</a>></span> wrote:<br><div class="gmail_quote"><blockquote class="gmail_quote" style="border-left: 1px solid rgb(204, 204, 204); margin: 0pt 0pt 0pt 0.8ex; padding-left: 1ex;">
<br>
> From: Matthew Knepley<br>
<div class="im">><br>
> On Thu, Apr 16, 2009 at 11:34 AM, Chetan Jhurani <<a href="mailto:chetan@ices.utexas.edu">chetan@ices.utexas.edu</a>> wrote:<br>
><br>
</div><div class="im">> > Only a square matrix can be singular.<br>
><br>
> No, a singular matrix has a kernel. A non-square matrix can be singular.<br>
<br>
</div>One can generalize the concept of singular for rank-deficient rectangular<br>
matrices, but almost all usual definitions of singular matrix use<br>
non-invertibility or determinant and thus restrict themselves to<br>
square matrices.<br>
<br>
For example, <a href="http://mathworld.wolfram.com/SingularMatrix.html" target="_blank">http://mathworld.wolfram.com/SingularMatrix.html</a>.<br><div class="im"></div></blockquote><div><br>The definition that makes the most sense (and generalizes far beyond matrices)<br>
is |ker(A)| > 0.<br><br> Matt<br> </div><blockquote class="gmail_quote" style="border-left: 1px solid rgb(204, 204, 204); margin: 0pt 0pt 0pt 0.8ex; padding-left: 1ex;"><div class="im">
> > If rank(A) = n, see<br>
> > <<a href="http://en.wikipedia.org/wiki/Moore-Penrose_pseudoinverse#The_QR_method" target="_blank">http://en.wikipedia.org/wiki/Moore-Penrose_pseudoinverse#The_QR_method</a>><br>
><br>
> QR will work for a matrix of rank < n. In this case, a null space basis fills out U.<br>
<br>
</div>Agreed.<br>
<font color="#888888"><br>
Chetan<br>
<br>
</font></blockquote></div><br><br clear="all"><br>-- <br>What most experimenters take for granted before they begin their experiments is infinitely more interesting than any results to which their experiments lead.<br>-- Norbert Wiener<br>