spectrum of Hadamard matrices

For a symmetric Hadamard matrix $H$ of order $m$ we have the minimal polynomial $x^2-m$, i.e. eigenvalues $\pm\sqrt{m}$. Indeed, by Cayley-Hamilton theorem $HH^\top=H^2=mI$, as $HH^\top=mI$ for any Hadamard matrix.

In general, eigenvalues are not $\pm\sqrt{m}$. E.g. if I take a skew-symmetric Hadamard matrix of order 4, I get minimal polynomial $x^2-2x+4$.So you get complex eigenvalues. Actually, for any skew-symmetric order $m$ Hadamard matrix $H$ with constant diagonal (i.e. $(H-I)^\top=-H+I$) the minimal polynomial is $p(x)=x^2-2x+m$. Indeed, one gets $H=2I-H^\top$, and plugging this into $p(x)$ gives $H(2I-H^\top)-2H+mI=0$, as required.

The evidence from Sagemath was helpful:

 sage: from sage.combinat.matrices.hadamard_matrix import skew_hadamard_matrix
 sage: skew_hadamard_matrix(4).minimal_polynomial()
 x^2 - 2*x + 4
 sage: skew_hadamard_matrix(8).minimal_polynomial()
 x^2 - 2*x + 8
 sage: skew_hadamard_matrix(12).minimal_polynomial()
 x^2 - 2*x + 12
 sage: skew_hadamard_matrix(16).minimal_polynomial()
 x^2 - 2*x + 16