Prove that the Hadamard matrix manifold

is in general not smooth, and nor it is a complex algebraic manifold.

Prove that the dephased Hadamard matrix manifold

is in general not smooth, and not a complex algebraic manifold either.

Prove that the set [math]E_N[/math] formed by the [math]N\times N[/math] complex Hadamard matrices modulo the equivalence relation is given by

and compute this set at [math]N=2,3,4,5[/math].

Work out the formula of the dephased defect of the Fourier matrix [math]F_N[/math], and then of the generalized Fourier matrix [math]F_G[/math].

Find an alternative proof for the formula

for the real Hadamard matrices, [math]H\in M_N(\pm1)[/math].

Find the defect of the following matrix,

via the simplest possible proof.

Prove that the Tao matrix,

with [math]w=e^{2\pi i/3}[/math], is isolated in the dephased Hadamard matrix manifold.

Is the defect always equal to the number of [math]1[/math] entries?

Prove that given two Hadamard matrices [math]H,K[/math], we have:

Is this actually always an equality, or not?

Develop a defect theory for the partial Hadamard matrices

notably by finding the defect equations, in this setting.