Comparing eigenvalues of two matrices - MathOverflow [closed]most recent 30 from http://mathoverflow.net2013-06-19T15:20:19Zhttp://mathoverflow.net/feeds/question/109098http://www.creativecommons.org/licenses/by-nc/2.5/rdfhttp://mathoverflow.net/questions/109098/comparing-eigenvalues-of-two-matricesComparing eigenvalues of two matriceshayu2012-10-07T20:29:13Z2012-10-07T20:29:13Z
<p>Suppose we have</p>
<p>$A=\left(\begin{array}{cccc}
1 & 1 & 1 & 0\\
1 & 3 & 0 & 0\\
1 & 0 & 2 & 1\\
0 & 0 & 1 & 3
\end{array}\right)$</p>
<p>and</p>
<p>$B=\left(\begin{array}{cccc}
2 & 1 & 0 & 0\\
1 & 1 & 1 & 1\\
0 & 1 & 3 & 0\\
0 & 1 & 0 & 3
\end{array}\right)$</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>