Codes and Expansions (CodEx) Seminar

Chris Manon (University of Kentucky)
Spaces of Eigenvalues

For an \(m \times n\) matrix \(\Phi\) with columns \(\phi_1, \ldots, \phi_n\) and a subset \(A \subset [n]\) of the index set of the columns, let \(\lambda_A(\Phi)\) be the spectrum of the Hermitian matrix \(\sum_{i \in A} \phi_i\phi_i^*.\) For a set of such sets \(S = \{A_1, \ldots, A_k\}\) we get a map \(\lambda_S: M_{m \times n}(\mathbb{C}) \to (\mathbb{R}^m)^{|S|}\). We call the image \(P_S\) of \(\lambda_S\) a space of eigenvalues. In this talk I'll explain how a few familiar objects from representation theory and geometry appear as \(P_S\) for some set \(S\), give a general condition implying that \(P_S\) is a convex polyhedral cone, and give a general condition implying that \(P_S\) is not convex. This is joint work with Alex Fink and Milena Hering.