Comparing eigenvalues of two matrices - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-06-19T15:20:19Z http://mathoverflow.net/feeds/question/109098 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/109098/comparing-eigenvalues-of-two-matrices Comparing eigenvalues of two matrices hayu 2012-10-07T20:29:13Z 2012-10-07T20:29:13Z <p>Suppose we have</p> <p>$A=\left(\begin{array}{cccc} 1 &amp; 1 &amp; 1 &amp; 0\\ 1 &amp; 3 &amp; 0 &amp; 0\\ 1 &amp; 0 &amp; 2 &amp; 1\\ 0 &amp; 0 &amp; 1 &amp; 3 \end{array}\right)$</p> <p>and</p> <p>$B=\left(\begin{array}{cccc} 2 &amp; 1 &amp; 0 &amp; 0\\ 1 &amp; 1 &amp; 1 &amp; 1\\ 0 &amp; 1 &amp; 3 &amp; 0\\ 0 &amp; 1 &amp; 0 &amp; 3 \end{array}\right)$</p> <p>Hence, both A and B has the same determinant, same trace, and same row (column) sum. How can we conclude that</p> <p>$\lambda_{1}(B)>\lambda_{1}(A)$</p> <p>without explicitly compute the eigenvalue or the eigenvector.</p> <p>where: $\lambda_{1}$ : the largest eigenvalue </p>