Spectra of Convex Hulls of Matrix Groups
| Autor: | Jankowski, Eric, Johnson, Charles R., Lim, Derek |
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| Rok vydání: | 2019 |
| Předmět: | |
| Zdroj: | Linear Algebra and its Applications 593 (2020) 74-89 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1016/j.laa.2020.01.018 |
| Popis: | 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. Comment: 19 pages. This work was completed at the 2019 Matrix Analysis REU at the College of William & Mary |
| Databáze: | arXiv |
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