# 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.