Generalizations of Oppenheim's inequality - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T04:55:27Z http://mathoverflow.net/feeds/question/105745 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/105745/generalizations-of-oppenheims-inequality Generalizations of Oppenheim's inequality Felix Goldberg 2012-08-28T18:13:12Z 2013-02-05T22:19:06Z <p>The well-known Oppenheim inequality says that for two positive definite matrices $A,B$ it holds that $\det(A \circ B) \geq (\prod{a_{ii}})\det(B)$. </p> <p>There has been a lot of beautiful work done extending it to cases when $A$ or $B$ or both of them are $M$-matrices or their inverses, or totally nonnegative. </p> <p>My question is: do you know of other extensions, in which $A$ is non-symmetric in an "interesting" way?</p> http://mathoverflow.net/questions/105745/generalizations-of-oppenheims-inequality/105796#105796 Answer by Denis Serre for Generalizations of Oppenheim's inequality Denis Serre 2012-08-29T06:57:37Z 2012-08-29T06:57:37Z <p>This is not a generalization to other matrix classes, but a generalization of the inequality, within the same class of Hermitian positive definite (or semi-definite) matrices. The flaw of Oppenheim's inequality is that the right-hand side is not symmetric in $A$ and $B$, unlike the left-hand side. Instead, S. Fallat &amp; C. Johnson proved a symmetric form of OI: $$\det(A \circ B)+\det A\det B \geq (\prod{a_{ii}})\det(B)+(\prod{b_{ii}})\det(A).$$ See Exercise 285 in my <a href="http://www.umpa.ens-lyon.fr/~serre/DPF/exobis.pdf" rel="nofollow">List of exercises</a> on Matrices.</p> http://mathoverflow.net/questions/105745/generalizations-of-oppenheims-inequality/106471#106471 Answer by Betrand for Generalizations of Oppenheim's inequality Betrand 2012-09-06T01:16:39Z 2012-09-06T01:16:39Z <p>This is not a generalization to other matrix classes, but replacing determinant by permanent. Actually, it is a conjecture made by Bapat and Sunder: Under the same conditions $per(A \circ B) \leq (\prod{a_{ii}})per(B)$.</p> <p>...but the following result due to Jiao [On a conjecture of H. Minc, Linear and Multilinear Algebra 32 (1992) 103–105.] couldn't surprise me more $$per(A \circ B)+per (A) per (B) \geq (\prod{a_{ii}})per(B)+(\prod{b_{ii}})per(A).$$</p>