Characterizing the 6× 6 complex Hadamard matrices (CHMs) is an open problem in linear algebra and quantum information. In this paper, we investigate the eigenvalues and eigenvectors of CHMs. We show that any n× n CHM with dephased form has two constant eigenvalues ±√n and has two constant eigenvectors. We obtain the maximum numbers of identical eigenvalues of 6× 6 CHMs with dephased form and we extend this result to arbitrary dimension. We also show that there is no 6× 6 CHM with four identical eigenvalues. We conjecture that the eigenvalues and eigenvectors of 6× 6 CHMs will lead to the complete classification of 6× 6 CHMs.
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Liang et al. (2024) studied this question.
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