most recent 30 from http://mathoverflow.net 2013-05-22T22:15:14Z http://mathoverflow.net/feeds/question/53090 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/53090/eigenspace-of-sum-of-a-non-symmetric-matrix-and-its-transpose Abhishek Kumar 2011-01-24T17:18:25Z 2011-01-25T06:18:09Z

<p>Suppose $A$ is a non-symmetric matrix (also, not a normal matrix) with all non-negative eigenvalues. Is there a relation between eigenspace (subspace spanned by eigenvectors) of $A$ and eigenspace of $(A+A^T)$? Is there an overlap? One obvious observation is that row space of $A$ is same as column space of $A^T$. </p>

http://mathoverflow.net/questions/53090/eigenspace-of-sum-of-a-non-symmetric-matrix-and-its-transpose/53199#53199 ARupinski 2011-01-25T06:18:09Z 2011-01-25T06:18:09Z

<p>Trivially one has that the rank of $A+A^t$ cannot be larger than 2*rank($A$) since rank($A$) = rank($A^t$).</p> <p>Without knowing additional constraints on $A$, I don't see that much can be said about the eigenvalues of $A+A^t$. For example, let $A = [[2,2],[6,k]]$, it is easy to check that $A$ is not normal for any value of $k$.</p> <p>Consider the cases $k=7,8,9$. In each case $A$ has 2 positive eigenvalues, but:</p> <p>For k=7, $A+A^t$ has 1 positive, 1 negative eigenvalue</p> <p>For k=8, $A+A^t$ has one positive and one zero eigenvalue</p> <p>For k=9, $A+A^t$ has 2 positive eigenvalues</p> <p>So for $k\neq 8$, $A+A^t$ has the same eigenspace as $A$ (although the eigenvectors are different) while when $k=8$, the eigenspace of $A+A^t$ is a strict subspace of the eigenspace of $A$.</p>