Hermitian matrices of roots of unity and their characteristic polynomials
We investigate spectral conditions on Hermitian matrices of roots of unity. Our main results are conjecturally sharp upper bounds on the number of residue classes of the characteristic polynomial of such matrices modulo ideals generated by powers of (1−ζ), where ζ is a root of unity. We also prove a...
| Main Authors: | , |
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| Other Authors: | |
| Format: | Journal Article |
| Language: | English |
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2023
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| Online Access: | https://hdl.handle.net/10356/170926 |
| _version_ | 1826121506879438848 |
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| author | Greaves, Gary Royden Watson Woo, Chin Jian |
| author2 | School of Physical and Mathematical Sciences |
| author_facet | School of Physical and Mathematical Sciences Greaves, Gary Royden Watson Woo, Chin Jian |
| author_sort | Greaves, Gary Royden Watson |
| collection | NTU |
| description | We investigate spectral conditions on Hermitian matrices of roots of unity. Our main results are conjecturally sharp upper bounds on the number of residue classes of the characteristic polynomial of such matrices modulo ideals generated by powers of (1−ζ), where ζ is a root of unity. We also prove a generalisation of a classical result of Harary and Schwenk about a relation for traces of powers of a graph-adjacency matrix, which is a crucial ingredient for the proofs of our main results. |
| first_indexed | 2024-10-01T05:33:42Z |
| format | Journal Article |
| id | ntu-10356/170926 |
| institution | Nanyang Technological University |
| language | English |
| last_indexed | 2024-10-01T05:33:42Z |
| publishDate | 2023 |
| record_format | dspace |
| spelling | ntu-10356/1709262023-10-09T02:17:57Z Hermitian matrices of roots of unity and their characteristic polynomials Greaves, Gary Royden Watson Woo, Chin Jian School of Physical and Mathematical Sciences Science::Mathematics Hermitian Matrices Roots of Unity We investigate spectral conditions on Hermitian matrices of roots of unity. Our main results are conjecturally sharp upper bounds on the number of residue classes of the characteristic polynomial of such matrices modulo ideals generated by powers of (1−ζ), where ζ is a root of unity. We also prove a generalisation of a classical result of Harary and Schwenk about a relation for traces of powers of a graph-adjacency matrix, which is a crucial ingredient for the proofs of our main results. Ministry of Education (MOE) The first author was supported in part by the Singapore Ministry of Education Academic Research Fund (Tier 1); grant numbers: RG21/20 and RG23/20. 2023-10-09T02:17:56Z 2023-10-09T02:17:56Z 2023 Journal Article Greaves, G. R. W. & Woo, C. J. (2023). Hermitian matrices of roots of unity and their characteristic polynomials. Journal of Combinatorial Theory, Series A, 200, 105793-. https://dx.doi.org/10.1016/j.jcta.2023.105793 0097-3165 https://hdl.handle.net/10356/170926 10.1016/j.jcta.2023.105793 2-s2.0-85165417577 200 105793 en RG21/20 RG23/20 Journal of Combinatorial Theory, Series A © 2023 Elsevier Inc. All rights reserved. |
| spellingShingle | Science::Mathematics Hermitian Matrices Roots of Unity Greaves, Gary Royden Watson Woo, Chin Jian Hermitian matrices of roots of unity and their characteristic polynomials |
| title | Hermitian matrices of roots of unity and their characteristic polynomials |
| title_full | Hermitian matrices of roots of unity and their characteristic polynomials |
| title_fullStr | Hermitian matrices of roots of unity and their characteristic polynomials |
| title_full_unstemmed | Hermitian matrices of roots of unity and their characteristic polynomials |
| title_short | Hermitian matrices of roots of unity and their characteristic polynomials |
| title_sort | hermitian matrices of roots of unity and their characteristic polynomials |
| topic | Science::Mathematics Hermitian Matrices Roots of Unity |
| url | https://hdl.handle.net/10356/170926 |
| work_keys_str_mv | AT greavesgaryroydenwatson hermitianmatricesofrootsofunityandtheircharacteristicpolynomials AT woochinjian hermitianmatricesofrootsofunityandtheircharacteristicpolynomials |