eigenspace of sum of a non-symmetric matrix and its transpose - MathOverflow 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 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 Answer by ARupinski for eigenspace of sum of a non-symmetric matrix and its transpose 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>