2019/09/30 by Eric Jankowski, Charles R. Johnson, Derek Lim
Computer Science · Mathematics · #Combinatorics #Composite material #Convex analysis #Convex combination #Convex hull #Convex optimization #Convex polytope #Geometry #Hull #Materials science #Mathematics #Mathematics and Applications #Matrix (chemical analysis) #Matrix Theory and Algorithms #Pure mathematics #Regular polygon #math.RT #math.SP
paper · pdf · doi:10.1016/j.laa.2020.01.018
published as Linear Algebra and its Applications 593 (2020) 74-89 · 19 pages. This work was completed at the 2019 Matrix Analysis REU at the College of William & Mary
arxiv created 2019/12/27 · openalex publication_date 2020/01/29 · arxiv updated 2020/04/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
The still-unsolved problem of determining the set of eigenvalues realized by n-by-n doubly stochastic matrices, those matrices with row sums and column sums equal to 1, has attracted much attention in the last century. This problem is somewhat algebraic in nature, due to a result of Birkhoff demonstrating that the set of doubly stochastic matrices is the convex hull of the permutation matrices. Here we are interested in a general matrix group G ⊆ GLn(ℂ) and the hull spectrum HS(G) of eigenvalues realized by convex combinations of elements of G. We show that hull spectra of matrix groups share many nice properties. Moreover, we give bounds on the hull spectra of matrix groups, determine HS(G) exactly for important classes of matrix groups, and study the hull spectra of representations of abstract groups.