Geodesic
New duality operator for complex circulant matrices and a conjecture of Ryser
Luis H. Gallardo1
1University of Brest
The electronic journal of combinatorics, Tome 23 (2016) no. 1
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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.
DOI : 10.37236/5237
Classification : 15B34
Mots-clés : Fourier matrix, Fourier transform, circulant Hadamard matrices, Ryser's conjecture

Luis H. Gallardo  1

1 University of Brest 
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     author = {Luis H. Gallardo},
     title = {New duality operator for complex circulant matrices and a conjecture of {Ryser}},
     journal = {The electronic journal of combinatorics},
     year = {2016},
     volume = {23},
     number = {1},
     doi = {10.37236/5237},
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Luis H. Gallardo. New duality operator for complex circulant matrices and a conjecture of Ryser. The electronic journal of combinatorics, Tome 23 (2016) no. 1. doi: 10.37236/5237

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