New Duality Operator for Complex Circulant Matrices and a Conjecture of Ryser
| Autor: | Luis H. Gallardo |
|---|---|
| Rok vydání: | 2016 |
| Předmět: |
Discrete mathematics
Applied Mathematics Order (ring theory) Duality (optimization) 010103 numerical & computational mathematics 0102 computer and information sciences Absolute value (algebra) 01 natural sciences Theoretical Computer Science Combinatorics Computational Theory and Mathematics 010201 computation theory & mathematics Complex Hadamard matrix Hadamard transform Discrete Mathematics and Combinatorics Geometry and Topology 0101 mathematics Circulant matrix Hadamard matrix Eigenvalues and eigenvectors Mathematics |
| Zdroj: | The Electronic Journal of Combinatorics. 23 |
| ISSN: | 1077-8926 |
| DOI: | 10.37236/5237 |
| Popis: | We associate to any given circulant complex matrix $C$ another one $E(C)$ such that $E(E(C)) = C^{*}$ the transpose conjugate of $C.$ All circulant Hadamard matrices of order $4$ satisfy a condition $C_4$ on their eigenvalues, namely, the absolute value of the sum of all eigenvalues is bounded above by $2.$ We prove by a "descent" that uses our operator $E$ that the only circulant Hadamard matrices of order $n \geq 4$, that satisfy a condition $C_n$ that generalizes the condition $C_4$ and that consist of a list of $n/4$ inequalities for the absolute value of some sums of eigenvalues of $H$ are the known ones. |
| Databáze: | OpenAIRE |
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