Title: Determinants of Sums of Normal Matrices

 

Abstract: 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.

 

This is joint work with Luke Oeding.