# Almost Hadamard matrices with complex entries

## TL;DR

This paper extends the concept of almost Hadamard matrices to complex entries, finding that such matrices are likely limited to the standard Hadamard matrices, with implications for the broader study of complex Hadamard matrices.

## Contribution

It introduces a complex extension of almost Hadamard matrices and provides evidence supporting the conjecture that only standard Hadamard matrices exist in this class.

## Key findings

- The conjecture that no non-trivial complex almost Hadamard matrices exist.
- Verification of the conjecture for known real almost Hadamard matrices and some complex extensions.
- Discussion of potential applications to complex Hadamard matrix research.

## Abstract

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.