We discuss an extension of the almost Hadamard matrix formalism, to the case of complex matrices. Quite surprisingly, the situation here is very different from the one in the real case, and our conjectural conclusion is that there should be no such matrices, besides the usual Hadamard ones. We verify this conjecture in a number of situations, and notably for most of the known examples of real almost Hadamard matrices, and for some of their complex extensions. We discuss as well some potential applications of our conjecture, to the general study of complex Hadamard matrices.
References
T. Banica, B. Collins and J.-M. Schlenker, On orthogonal matrices maximizing the 1-norm, Indiana Univ. Math. J. 59 (2010), 839–856. 1228.15013 10.1512/iumj.2010.59.3926T. Banica, B. Collins and J.-M. Schlenker, On orthogonal matrices maximizing the 1-norm, Indiana Univ. Math. J. 59 (2010), 839–856. 1228.15013 10.1512/iumj.2010.59.3926
T. Banica and I. Nechita, Almost Hadamard matrices: the case of arbitrary exponents, Discrete Appl. Math. 161 (2013), 2367–2379. 1285.05023 10.1016/j.dam.2013.05.012T. Banica and I. Nechita, Almost Hadamard matrices: the case of arbitrary exponents, Discrete Appl. Math. 161 (2013), 2367–2379. 1285.05023 10.1016/j.dam.2013.05.012
T. Banica, I. Nechita and J.-M. Schlenker, Submatrices of Hadamard matrices: complementation results, Electron. J. Linear Algebra 27 (2014), 197–212. MR3194951 1321.15051 10.13001/1081-3810.1613T. Banica, I. Nechita and J.-M. Schlenker, Submatrices of Hadamard matrices: complementation results, Electron. J. Linear Algebra 27 (2014), 197–212. MR3194951 1321.15051 10.13001/1081-3810.1613
G. Björck, Functions of modulus $1$ on ${\rm Z}_n$ whose Fourier transforms have constant modulus, and cyclic $n$-roots, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 315 (1990), 131–140. G. Björck, Functions of modulus $1$ on ${\rm Z}_n$ whose Fourier transforms have constant modulus, and cyclic $n$-roots, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 315 (1990), 131–140.
A. T. Butson, Generalized Hadamard matrices, Proc. Amer. Math. Soc. 13 (1962), 894–898. 0109.24605 10.1090/S0002-9939-1962-0142557-0A. T. Butson, Generalized Hadamard matrices, Proc. Amer. Math. Soc. 13 (1962), 894–898. 0109.24605 10.1090/S0002-9939-1962-0142557-0
C. J. Colbourn and J. H. Dinitz, Handbook of combinatorial designs, Second edition. Discrete Mathematics and its Applications (Boca Raton). Chapman & Hall/CRC, Boca Raton, FL, 2007. 1101.05001C. J. Colbourn and J. H. Dinitz, Handbook of combinatorial designs, Second edition. Discrete Mathematics and its Applications (Boca Raton). Chapman & Hall/CRC, Boca Raton, FL, 2007. 1101.05001
U. Haagerup, Orthogonal maximal abelian $*$-subalgebras of the $n\times n$ matrices and cyclic $n$-roots, in “Operator algebras and quantum field theory”, International Press (1997), 296–323. U. Haagerup, Orthogonal maximal abelian $*$-subalgebras of the $n\times n$ matrices and cyclic $n$-roots, in “Operator algebras and quantum field theory”, International Press (1997), 296–323.
K.J. Horadam, Hadamard matrices and their applications, Princeton University Press, Princeton, NJ, 2007. 1145.05014K.J. Horadam, Hadamard matrices and their applications, Princeton University Press, Princeton, NJ, 2007. 1145.05014
M. Idel and M. M. Wolf, Sinkhorn normal form for unitary matrices, Linear Algebra Appl. 471 (2015), 76–84. 1307.15014 10.1016/j.laa.2014.12.031M. Idel and M. M. Wolf, Sinkhorn normal form for unitary matrices, Linear Algebra Appl. 471 (2015), 76–84. 1307.15014 10.1016/j.laa.2014.12.031
B.R. Karlsson, Three-parameter complex Hadamard matrices of order 6, Linear Algebra Appl. 434 (2011), 247–258. 1231.05045 10.1016/j.laa.2010.08.020B.R. Karlsson, Three-parameter complex Hadamard matrices of order 6, Linear Algebra Appl. 434 (2011), 247–258. 1231.05045 10.1016/j.laa.2010.08.020
H. Kharaghani and B. Tayfeh-Rezaie, A Hadamard matrix of order 428, J. Combin. Des. 13 (2005), 435–440. 1076.05017 10.1002/jcd.20043H. Kharaghani and B. Tayfeh-Rezaie, A Hadamard matrix of order 428, J. Combin. Des. 13 (2005), 435–440. 1076.05017 10.1002/jcd.20043
W. de Launey and D. A. Levin, A Fourier-analytic approach to counting partial Hadamard matrices, Cryptogr. Commun. 2 (2010), 307–334. MR2719847 1225.05056 10.1007/s12095-010-0033-zW. de Launey and D. A. Levin, A Fourier-analytic approach to counting partial Hadamard matrices, Cryptogr. Commun. 2 (2010), 307–334. MR2719847 1225.05056 10.1007/s12095-010-0033-z
K. H. Leung and B. Schmidt, New restrictions on possible orders of circulant Hadamard matrices, Des. Codes Cryptogr. 64 (2012), 143–151. 1242.15027 10.1007/s10623-011-9493-1K. H. Leung and B. Schmidt, New restrictions on possible orders of circulant Hadamard matrices, Des. Codes Cryptogr. 64 (2012), 143–151. 1242.15027 10.1007/s10623-011-9493-1
H. J. Ryser, Combinatorial mathematics, The Carus Mathematical Monographs, No. 14 Published by The Mathematical Association of America; distributed by John Wiley and Sons, Inc., New York 1963. H. J. Ryser, Combinatorial mathematics, The Carus Mathematical Monographs, No. 14 Published by The Mathematical Association of America; distributed by John Wiley and Sons, Inc., New York 1963.
D. R. Stinson, Combinatorial designs: constructions and analysis, With a foreword by Charles J. Colbourn. Springer-Verlag, New York, 2004. 1031.05001D. R. Stinson, Combinatorial designs: constructions and analysis, With a foreword by Charles J. Colbourn. Springer-Verlag, New York, 2004. 1031.05001
J. J. Sylvester, Thoughts on inverse orthogonal matrices, simultaneous sign-successions, and tesselated pavements in two or more colours, with applications to Newton's rule, ornamental tile-work, and the theory of numbers, Phil. Mag. 34 (1867), 461–475. J. J. Sylvester, Thoughts on inverse orthogonal matrices, simultaneous sign-successions, and tesselated pavements in two or more colours, with applications to Newton's rule, ornamental tile-work, and the theory of numbers, Phil. Mag. 34 (1867), 461–475.
