Eigenvalues and eigenvectors of complex Hadamard matrices

Mengfan Liang; Lin Chen

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arXiv · 2408.10471

Eigenvalues and eigenvectors of complex Hadamard matrices

Mengfan Liang·Lin Chen

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Abstract

Characterizing the $6\times 6$ 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 $n\times n$ CHM with dephased form has two constant eigenvalues $\pm\sqrt{n}$ and has two constant eigenvectors. We obtain the maximum numbers of identical eigenvalues of $6\times 6$ CHMs with dephased form and we extend this result to arbitrary dimension. We also show that there is no $6\times 6$ CHM with four identical eigenvalues. We conjecture that the eigenvalues and eigenvectors of $6\times 6$ CHMs will lead to the complete classification of $6\times 6$ CHMs.

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Mengfan Liang, Lin Chen. 2024-08-20. Eigenvalues and eigenvectors of complex Hadamard matrices. https://arxiv.org/abs/2408.10471

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