Determinants of Sums of Normal Matrices

Algebra Seminar
Monday, January 30, 2023 - 10:20am for 1.5 hours (actually 80 minutes)
Skiles 005
Luke Oeding – Auburn University – oeding@auburn.edu
Abeer Al Ahmadieh

Marcus (1972) and de Oliveira (1982) conjectured  bounds on the determinantal range of the sum of a pair of normal matrices with prescribed eigenvalues.  We show that this determinantal range is a flattened solid twisted permutahedron, which is, in turn, a finite union of flattened solid twisted hypercubes with prescribed vertices.  This complete geometric description, in particular, proves the conjecture. Our techniques are based on classical Lie theory, geometry, and combinatorics. I will give a pre-seminar that will be accessible to 1st year graduate students who like matrices, and provides an easy introduction to the topic. This is joint work with Matt Speck.