bilinear equation OR diagonal matrix search - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T07:10:30Z http://mathoverflow.net/feeds/question/93166 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/93166/bilinear-equation-or-diagonal-matrix-search bilinear equation OR diagonal matrix search Boris 2012-04-04T20:48:17Z 2012-04-05T13:25:12Z <p>Dear guys, I am parametrizing a model and I face an interesting (but tough for me) problem</p> <p>I have a real square $n \times n$ symmetric matrix $B$ (which consists of 2 square blocks of positive numbers and 2 rectangular blocks of negative numbers) and I need a real diagonal matrix $D$ such that column vector of $n$ $1$'s ("ones(n,1)" in matlab syntax) is an eigenvector of $DBD$.</p> <p>In other words I need a vector $d$ for which $\sum_{j=1}^n B_{ij} d_i d_j = 1$ for $i=1..n$ ($B_{ij}$ are elements of the same matrix $B$)</p> <p>Thanks, very much.</p> <p>Boris</p> http://mathoverflow.net/questions/93166/bilinear-equation-or-diagonal-matrix-search/93205#93205 Answer by Felix Goldberg for bilinear equation OR diagonal matrix search Felix Goldberg 2012-04-05T09:23:33Z 2012-04-05T09:52:51Z <p>$DBD$ is in effect a re-scaling of the rows and columns of $B$, and you want it to have constant row-sums. If $B$ were positive, this would be possible by, say, Sinkhorn's theorem. For $B$ with negative values, this would be much more difficult, but possible in special cases. See this survey by C.R.Johnson &amp; R.Reams for a panorama of the subject:</p> <p><strong>Scaling of Symmetric Matrices by Positive Diagonal Congruence</strong> <a href="http://faculty.plattsburgh.edu/robert.reams/research/sinkhorn.pdf" rel="nofollow">http://faculty.plattsburgh.edu/robert.reams/research/sinkhorn.pdf</a> </p> <p>P.S. Do you have more detailed information about the structure of your matrix $B$?</p> http://mathoverflow.net/questions/93166/bilinear-equation-or-diagonal-matrix-search/93214#93214 Answer by Ilya Bogdanov for bilinear equation OR diagonal matrix search Ilya Bogdanov 2012-04-05T13:25:12Z 2012-04-05T13:25:12Z <p>As far as I understand, you can multiply your matrix from both sides by the matrix $\left(\matrix{E_p&amp;0\cr 0&amp;-E_r}\right)$ with a suitable sizes of unit matrices in order to obtain a matrix $B'$ with positive entries. Then Sinkhorn's theorem is applicable, as Felix mentioned.</p> <p>(Surely, this theorem provides TWO diagonal matrices $D_1$, $D_2$ with positive numbers on the diagonal such that $D_1B'D_2$ is doubly stochastic. But, since these matrices are unique up to the scaling, they should coincide up to a scalar factor.)</p>