http://openproblemgarden.org/op/on_gersgorin_theorem Comments for "On Gersgorin Theorem" en http://openproblemgarden.org/op/on_gersgorin_theorem#comment-6685 <p>Take any diagonal matrix. Then the eigenvalues are the diagonal entries but the the Gerschgorin discs are points centred at the diagonal entries. So we can't remove one of the discs from the set. </p> Wed, 18 Nov 2009 14:28:29 +0100 alext87 comment 6685 at http://openproblemgarden.org http://openproblemgarden.org/op/on_gersgorin_theorem#comment-6653 <p>Doesn't the matrix <img class="teximage" src="/files/tex/788584cded89b1495f6c7830f9aaee584d5606c2.png" alt="$ \begin{pmatrix} 1 &amp; 2 &lt;br&gt; 2 &amp; 2 \end{pmatrix} $" /> provide a counterexample?</p> Tue, 23 Jun 2009 00:11:33 +0200 Anonymous comment 6653 at http://openproblemgarden.org http://openproblemgarden.org/op/on_gersgorin_theorem#comment-2426 <p>It remains open.</p> Sun, 12 Oct 2008 11:41:20 +0200 Miwa Lin comment 2426 at http://openproblemgarden.org http://openproblemgarden.org/op/on_gersgorin_theorem <table cellspacing="10"> <tr> <td> Author(s): <a href="/kpz_equation_central_limit_theorem"></a>&nbsp;&nbsp; </td> <td align=right> Subject: <a href="/category/algebra">Algebra</a>&nbsp;&nbsp; </td> </tr> <tr> <td colspan=2> <table border=1 cellspacing="5"> <tr><td> <p>Gersgorin theorem states that: all the eigenvalues of <img class="teximage" src="/files/tex/f871e0e7c9a9a6ce437fbea5a847f5b304447990.png" alt="$ A=[a_{ij}]\in M_n $" /> are located in the union of <img class="teximage" src="/files/tex/ec63d7020a64c039d5f6703b8fa3ab7393358b5b.png" alt="$ n $" /> discs <img class="teximage" src="/files/tex/36d726d49e61924405e168c8aa2c5dcfe1c6d8b4.png" alt="$ \bigcup\limits_{i=1}^n\{z\in C:|z-a_{ii}|\leq \sum\limits_{j=1,j\neq i}^n|a_{ij}|\} $" />. For some special matrices, the region can be confined to <img class="teximage" src="/files/tex/4884c1cf888ddcd0e9a4eb850408c33fe30e2b65.png" alt="$ \bigcup\limits_{i=1}^n\{z\in C:|z-a_{ii}|\leq \sum\limits_{j=1,j\neq i}^n|a_{ij}|\}\backslash\{z\in C:|z-a_{kk}|&lt;\sum\limits_{j=1,j\neq k}^n|a_{kj}|\} $" /> for some <img class="teximage" src="/files/tex/c450c3185f7285cfa0b88d3a903c54f7df601201.png" alt="$ k $" />. I wonder if the new region above is valid in general?</p> </tr></td> </table> </td> </tr> </table> Algebra http://openproblemgarden.org/op/on_gersgorin_theorem#comment Sat, 11 Oct 2008 15:13:23 +0200 Miwa Lin 3032 at http://openproblemgarden.org