  We prove that if $A$ and $B$ are Hadamard matrices which are both of size $4
\times 4$ or $5 \times 5$ and in dephased form, then $tr(A) = tr(B)$ implies
that $A$ and $B$ have the same eigenvalues, including multiplicity. We
calculate explicitly the spectrum for these matrices. We also extend these
results to larger Hadamard matrices which are permutations of the Fourier
matrix and calculate their spectral multiplicities.
