Applied Math Colloquium: Muddappa Gowda

Abstract: We consider a system where the sharpened Cauchy-Schwarz inequality holds. Formally called a semi-FTvN system, this is a triple (V, W, λ), where V and W are real inner product spaces and the mapping λ : V → W satisfies the sharpened Cauchy-Schwarz inequality ⟨x, y⟩ ≤ ⟨λ(x), λ(y)⟩ ≤ ||x|| ||y|| (∀ x, y ∈ V ). Such a system arises from a complete hyperbolic system (where λ is the eigenvalue map) and is a generalization of the Fan-Theobald-von Neumann system (e.g., Euclidean Jordan algebra). Motivated by matrix theory and optimization considerations, in this talk, we introduce two commutativity concepts: Strong commutativity via the equality ⟨x, y⟩ = ⟨λ(x), λ(y)⟩, and commutativity via the condition ⟨Dx, y⟩ = 0 for all D in the Lie algebra of the automorphism group of the system. We show that these reduce to the familiar commutativity concepts in the space of real symmetric matrices and Euclidean Jordan algebras. In the form of applications, we describe several commutation principles (where commutativity shows up as an optimality condition) for optimization problems over matrix Lie groups and for variational inequality problems.