Thanks Prof. Sun's beautiful conjecture! I have obtained a proof for (4) by multiplying the Hermitian matrix by a diagonal matrix with diagonal entries all equal to the product of the non-trivial n-1 roots. Then it is not hard to show that the resulting matrix has eigenvalues with corresponding eigenvectors in the form of (exp(s2pii/n))_j for s=-(n-1)/2,...-1,1,(n-1)/2.
Keqin Liu 'Kevin'
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