Decomposition of a Hermitian matrix into a sum of a fixed number of orthoprojections
We prove that any Hermitian matrix, whose trace is integer and all eigenvalues lie in $[1+1/(k-3),k-1-1/(k-3)],$ is a sum of $k$ orthoprojections. For sums of $k$ orthoprojections, it is shown that the ratio of the number of eigenvalues not exceeding 1 to the number of eigenvalues not less than 1, t...
Saved in:
| Date: | 2020 |
|---|---|
| Main Authors: | , |
| Format: | Article |
| Language: | Ukrainian |
| Published: |
Institute of Mathematics, NAS of Ukraine
2020
|
| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/2378 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| Summary: | We prove that any Hermitian matrix, whose trace is integer and all eigenvalues lie in $[1+1/(k-3),k-1-1/(k-3)],$ is a sum of $k$ orthoprojections. For sums of $k$ orthoprojections, it is shown that the ratio of the number of eigenvalues not exceeding 1 to the number of eigenvalues not less than 1, taking into account the multiplicity, is not greater than $k-1$. Examples of Hermitian matrices that satisfy the ratio for eigenvalues and, at the same time, can not be decomposed into a sum of $k$ orthoprojections are also suggested. |
|---|---|
| DOI: | 10.37863/umzh.v72i5.2378 |