Spectral properties of hermitean matrices whose entries are roots of unity
Let H_n(q) denote the set of all n by n Hermitean matrices whose entries are qth roots of unity. This thesis studies the spectral properties of matrices in H_n(q) for n, q in natural number N. We determine (conjecturally sharp) upper bounds for the number of residue classes of characteristic poly...
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| Format: | Thesis-Doctor of Philosophy |
| Language: | English |
| Published: |
Nanyang Technological University
2022
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| Online Access: | https://hdl.handle.net/10356/155732 |
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| Institution: | Nanyang Technological University |
| Language: | English |
| Summary: | Let H_n(q) denote the set of all n by n Hermitean matrices whose entries are qth roots
of unity. This thesis studies the spectral properties of matrices in H_n(q) for n, q in natural number N.
We determine (conjecturally sharp) upper bounds for the number of residue classes of
characteristic polynomials of matrices in H_n(q), modulo ideals generated by powers of
(1 - zeta), where zeta is a primitive qth root of unity. We prove a generalisation of a classical
result of Harary and Schwenk on a congruence of traces modulo ideal (1 - zeta ), which is a
crucial ingredient for the proofs of our main results. We also prove that, when n is odd,
the switching class of each matrix in H_n(q) contains exactly one Euler graph. Lastly,
we solve a problem of Et-Taoui about a potential sufficient condition for the switching
equivalence of Seidel matrices. |
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