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Spectra of Convex Hulls of Matrix Groups
- Publication Year :
- 2019
- Publisher :
- arXiv, 2019.
-
Abstract
- 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 $G \subseteq GL_n(\mathbb{C})$ and the hull spectrum $\text{HS}(G)$ 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 $\text{HS}(G)$ exactly for important classes of matrix groups, and study the hull spectra of representations of abstract groups.<br />Comment: 19 pages. This work was completed at the 2019 Matrix Analysis REU at the College of William & Mary
- Subjects :
- Convex hull
Numerical Analysis
Algebra and Number Theory
Group (mathematics)
010102 general mathematics
Spectrum (functional analysis)
010103 numerical & computational mathematics
Permutation matrix
01 natural sciences
Combinatorics
Mathematics - Spectral Theory
Matrix group
Hull
FOS: Mathematics
Discrete Mathematics and Combinatorics
Geometry and Topology
0101 mathematics
Algebraic number
Representation Theory (math.RT)
Spectral Theory (math.SP)
Eigenvalues and eigenvectors
Mathematics - Representation Theory
Mathematics
Subjects
Details
- Database :
- OpenAIRE
- Accession number :
- edsair.doi.dedup.....53bc8dd6908338340599d9da92296fac
- Full Text :
- https://doi.org/10.48550/arxiv.1909.10597