Almost Hadamard Matrices with Complex Entries

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. Introduction An Hadamard matrix is a square matrixH ∈MN(±1), whose rows are pairwise orthogonal. Here is a basic example, which appears as a version of the 4×4 Walsh matrix, and is also well-known for its use in the Grover algorithm: K4 =  −1 1 1 1 1 −1 1 1 1 1 −1 1 1 1 1 −1  Assuming that the matrix has N ≥ 3 rows, the orthogonality conditions between the rows give N ∈ 4N. A similar analysis with four or more rows, or any Copyright 2016 by the Tusi Mathematical Research Group. Date: Received: Feb. 9, 2017; Accepted: May 12, 2017. ∗Corresponding author. 2010 Mathematics Subject Classification. Primary 15B10; Secondary 05B20, 14P05.