Singular values of the sum of A and A^T - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T06:53:32Z http://mathoverflow.net/feeds/question/117567 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/117567/singular-values-of-the-sum-of-a-and-at Singular values of the sum of A and A^T Daniel86 2012-12-29T22:22:05Z 2013-01-03T06:24:26Z <p>Dear all,</p> <p>As a part of my research, I need to achieve a lower bound to the smallest singular value, $\sigma_{n}(A+A^{T})$ for a stochastic $A$ (as a function of the singular values of $A$), which is generally not Hermitian. </p> <p>I am aware of the upper bounds (due to Weyl and Fan) and of the fact that for general $\sigma_{i}(A+B)$ no lower bound is known. Do you see a way?</p> <p>Thank you.</p> <p>Edit: A can be considered a power of a lazy row stochastic matrix. I.e., $A=P^k$ for some strongly diagonally dominant row stochastic $P$.</p> http://mathoverflow.net/questions/117567/singular-values-of-the-sum-of-a-and-at/117913#117913 Answer by alex o. for Singular values of the sum of A and A^T alex o. 2013-01-02T22:43:49Z 2013-01-02T22:43:49Z <p>It does not seem like any obvious bound on $\sigma_n(A+A^T)$ is possible in terms of the singular values of $A$. Indeed, consider $$A = \left( \matrix{ 1/3 &amp; 1/2 &amp; 1/6 \cr 1/6 &amp; 1/3 &amp; 1/2 \cr 1/2 &amp; 1/6 &amp; 1/3} \right).$$ Clearly, $A$ is stochastic and a computation reveals that it is nonsingular so that all of its singular values are positive. On the other hand, $A+A^T$ is a multiple of the all-ones matrix, so $\sigma_3(A+A^T)=0$. </p>