Eigenvalues of Circulant Matrices and a Conjecture of Ryser

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DOI: 10.46793/KgJMat2105.751E


We prove that there is no circulant Hadamard matrix H with first row [h1,,hn] of order n > 4, under some linear conditions on the hi’s. All these conditions hold in the known case n = 4, so that our results can be thought as characterizations of properties that only hold when n = 4. Our first conditions imply that some eigenvalue λ of H is a sum of √ --
  n terms h jωj, where ω is a primitive n-th root of 1. The same conclusion holds also if some complex arithmetic means associated to λ are algebraic integers (second conditions). Moreover, our third conditions, related to the recent notion of robust Hadamard matrices, implies also the nonexistence of these circulant Hadamard matrices. If some of the conditions fail, it appears (to us) very difficult to be able to prove the result.


Circulant matrices, Hadamard matrices, eigenvalues, unit circle, cyclotomic fields.


[1]   P. Borwein and M. J. Mossinghoff, Wieferich pairs and Barker sequences II, LMS J. Comput. Math. 17(1) (2014), 24–32.

[2]   R. A. Brualdi, A note on multipliers of difference sets, Journal of Research of the National Bureau of Standards, Section B 69 (1965), 87–89.

[3]   P. J. Davis, Circulant Matrices, 2nd ed., AMS Chelsea Publishing, New York, 1994.

[4]   R. Euler, L. H. Gallardo and O. Rahavandrainy, Sufficient conditions for a conjecture of Ryser about Hadamard Circulant matrices, Linear Algebra Appl. 437 (2012), 2877–2886.

[5]   R. Euler, L. H. Gallardo and O. Rahavandrainy, Combinatorial properties of circulant Hadamard matrices, in: C. M. da Fonseca, D. Van Huynh, S. Kirkland and V. K. Tuan (Eds.), A panorama of Mathematics: Pure and Applied, Contemporary Mathematics (Book 658), American Mathematical Society, Providence, RI, 2016, 9–19.

[6]   A. G¸a    siorowski, G. Rajchel and K. Zyczkowski, Robust Hadamard matrices, unistochastic rays in Birkhoff polytope and equi-entangled bases in composite spaces, Math. Comput. Sci. 12(4) (2018), 473–490.

[7]   A. Hedayat and W. D. Wallis, Hadamard matrices and their applications, Ann. Statist. 6(6) (1978), 1184–1238.

[8]   J. Jedwab and S. Lloyd, A note on the nonexistence of Barker sequences, Des. Codes Cryptogr. 2(1) (1992), 93–97.

[9]   L. Gallardo, On a special case of a conjecture of Ryser about Hadamard circulant matrices, Appl. Math. E-Notes 12 (2012), 182–188.

[10]   L. H. Gallardo, New duality operator for complex circulant matrices and a conjecture of Ryser, Electron. J. Combin. 23(1) (2016), Paper ID 1.59, 10 pages.

[11]   M. Matolcsi, A Walsh-Fourier approach to the circulant Hadamard conjecture, in: Algebraic Design Theory and Hadamard Matrices, Springer Proc. Math. Stat. 133, Springer, Cham, 2015, 201–208.

[12]   D. B. Meisner, On a construction of regular Hadamard matrices, Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Serie IX. Matematica e Applicazioni 3(4) (1992), 233–240.

[13]   Y. Y. Ng, Cyclic Menon difference sets, circulant hadamard matrices and Barker sequences, Master Thesis, The University of Hong Kong, December 1993, 36 pages.

[14]   M. J. Mossinghoff, Wieferich prime pairs, Barker sequences, and circulant Hadamard matrices, 2013, [http://www.cecm.sfu.ca/ mjm/WieferichBarker/].

[15]   K. H. Leung and B. Schmidt, New restrictions on possible orders of circulant Hadamard matrices, Des. Codes Cryptogr. 64 (2012), 143–151.

[16]   H. J. Ryser, Combinatorial Mathematics, The Carus Mathematical Monographs 14, The Mathematical Association of America, John Wiley and Sons, Inc., New York, 1963.

[17]   J.-P. Serre, Finite Groups: An Introduction, Surveys of Modern Mathematics 10, International Press, Somerville, MA, Higher Education Press, Beijing, 2016.

[18]   R. J. Turyn, Character sums and difference sets, Pac. J. Math. 15 (1965), 319–346.