Optimization Seminar: Dr. M. Seetharama Gowda

In an Euclidean Jordan algebra V of rank n, an element x is said to be majorized by an element y, symbolically x ≺ y, if the corresponding eigenvalue vector λ(x) is majorized by λ(y) in Rⁿ. In this talk, we describe pointwise majorization inequalities of the form T(x) ≺ S(x), where T and S are linear transformations induced by Schur products. Specializing, we describe analogs of majorization inequalities of Schur, Hadamard, and Oppenheimer stated in the setting of Euclidean Jordan algebras, as well as majorization inequalities connecting quadratic and Lyapunov transformations on V . We also show how Schur products induced by certain scalar means (such as arithmetic, geometric, harmonic, and logarithmic means) naturally lead to majorization inequalities.