New Duality Operator for Complex Circulant Matrices and a Conjecture of Ryser
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.
1977 ◽
Vol 32
(11-12)
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pp. 908-912
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1995 ◽
Vol 28
(14)
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pp. 2847-2862
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1952 ◽
Vol 213
(1114)
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pp. 408-424
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1996 ◽
Vol 35
(6)
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pp. 830-838
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