Maximizing traces of matrix functions Original Research Article

Let X and Y be n×n Hermitian matrices with eigenvalues x1greater-or-equal, slantedx2greater-or-equal, slantedcdots, three dots, centeredgreater-or-equal, slantedxn and y1greater-or-equal, slantedy2greater-or-equal, slantedcdots, three dots, centeredgreater-or-equal, slantedyn respectively. We establish the inequalityimagetr(phi(X+Y))less-than-or-equals, slantmaxσset membership, variantSn∑j=1nphi(xj+yσ(j)),where Sn denotes the group of permutations and for phi satisfying a certain analytical condition. We establish also the inequalityimagetr(phi(AB))less-than-or-equals, slantmaxσset membership, variantSn∑j=1nphi(ajbσ(j))for A and B be positive definite n×n matrices with eigenvalues a1greater-or-equal, slanteda2greater-or-equal, slantedcdots, three dots, centeredgreater-or-equal, slantedan>0 and b1greater-or-equal, slantedb2greater-or-equal, slantedcdots, three dots, centeredgreater-or-equal, slantedbn>0 respectively and where tmaps tophi(et) satisfies the same analytical condition. As a consequence of the first of these inequalities, a conjecture of Drury, Liu, Lu, Puntanen and Styan concerning the sum of squares of canonical correlations is settled in the affirmative.