It is shown that a normalized complex Hadamard matrix of order 6 having three distinct columns each containing at least one \(-1\) entry, necessarily belongs to the transposed Fourier family, or to the family of 2-circulant complex Hadamard matrices. The proofs rely on solving polynomial systems of equations by Gröbner basis techniques, and make use of a structure theorem concerning regular Hadamard matrices. As a consequence, members of these two families can be easily recognized in practice. In particular, one can identify complex Hadamard matrices appearing in known triplets of pairwise mutually unbiased bases in dimension 6.

结果表明，一个 6 阶的归一化复数 Hadamard 矩阵具有三个不同的列，每个列至少包含一个 \（-1\） 条目，必然属于转置的傅里叶族，或者属于 2 循环复数 Hadamard 矩阵的族。这些证明依赖于通过 Gröbner 基技术求解多项式方程组，并利用有关正则 Hadamard 矩阵的结构定理。因此，这两个家族的成员在实践中很容易被识别。特别是，人们可以识别出现在维度 6 中成对互无偏基数的已知三元组中的复杂 Hadamard 矩阵。