Hölder type matrix inequalities of Pate, Blakley, and Roy extended to the inner product of Frobenius

Abstract

Suppose m, n, and k are positive integers, and let 〈·,·〉 be the standard inner product on the spaces Rp, p>0. Recently Pate has shown that if D is an m×n non-negative real matrix, and u and v are non-negative unit vectors in Rn and Rm, respectively, then〈(DDt)kDu,v〉⩾〈Du,v〉2k+1,with equality if and only if 〈(DDt)kDu,v〉=0, or there exists α>0 such that Du=αv and Dtv=αu. This extends to non-symmetric non-square matrices a 1965 result of Blakley and Roy, and resolves a special case of a graph theoretic inequality conjectured by Sidorenko. We generalize the above, obtaining pure matrix inequalities involving the Frobenius inner product, 〈·,·〉f. In particular, we show that if k is a positive integer, and D, X, and Y are non-negative matrices that are m×n,n×p, and m×p, respectively, then∑i=1p‖xi‖‖yi‖2k〈D(DtD)kX,Y〉f⩾(〈DX,Y〉f)2k+1,where X has columns x1,x2,…,xp, Y has columns y1,y2,…,yp, and ‖·‖ is the 2-norm. Necessary and sufficient conditions for equality are also given.