Generalizing the spectral radius of a unistochastic matrix - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T19:55:02Z http://mathoverflow.net/feeds/question/94078 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/94078/generalizing-the-spectral-radius-of-a-unistochastic-matrix Generalizing the spectral radius of a unistochastic matrix Victor Liu 2012-04-15T00:47:08Z 2012-04-15T09:18:10Z <p>Consider a square matrix \$A\$, and from it construct \$B\$ whose entries are the squared magnitudes of those in \$A\$. What can we say about the spectral radius of \$B\$? I know that for a unitary matrix \$A\$, \$B\$ is unistochastic so its spectral radius is 1, but I'm interested in the general case for arbitrary \$A\$.</p> <p>Also, relatedly, for general \$A\$, what can be said of the column 1-norms of \$B\$? And what about the the column 1-norms of products of such \$B\$-type matrices?</p> http://mathoverflow.net/questions/94078/generalizing-the-spectral-radius-of-a-unistochastic-matrix/94094#94094 Answer by Felix Goldberg for Generalizing the spectral radius of a unistochastic matrix Felix Goldberg 2012-04-15T08:30:41Z 2012-04-15T09:18:10Z <p>Well, \$B\$ is the Hadamard product of \$A\$ with itself, so if \$A\$ is nonnegative or psd, its radius is bounded from above by the square of the radius of \$A\$. </p> <p>For more information see this paper:</p> <p><a href="http://www.math.technion.ac.il/iic/ela/ela-articles/articles/vol20_pp90-94.pdf" rel="nofollow">http://www.math.technion.ac.il/iic/ela/ela-articles/articles/vol20_pp90-94.pdf</a></p>