Exercises

Apr 20'25

Prove that the Fourier matrices [math]F_2,F_3[/math], which are given by

[[math]] F_2=\begin{pmatrix}1&1\\ 1&-1\end{pmatrix}\quad,\quad F_3=\begin{pmatrix}1&1&1\\ 1&w&w^2\\ 1&w^2&w\end{pmatrix} [[/math]]

with [math]w=e^{2\pi i/3}[/math] are the only Hadamard matrices at [math]N=2,3[/math], up to equivalence.

Apr 20'25

If [math]H\in M_M(\mathbb T)[/math] and [math]K\in M_N(\mathbb T)[/math] are complex Hadamard matrices, prove that so is the matrix

[[math]] H\otimes_QK\in M_{MN}(\mathbb T) [[/math]]

given by the following formula, with [math]Q\in M_{M\times N}(\mathbb T)[/math],

[[math]] (H\otimes_QK)_{ia,jb}=Q_{ib}H_{ij}K_{ab} [[/math]]

called Di\c t\u a deformation of [math]H\otimes K[/math], with parameter [math]Q[/math].

Apr 20'25

Prove that the only complex Hadamard matrices at [math]N=4[/math] are, up to the standard equivalence relation, the matrices

[[math]] F_4^q=\begin{pmatrix} 1&1&1&1\\ 1&-1&1&-1\\ 1&q&-1&-q\\ 1&-q&-1&q \end{pmatrix} [[/math]]

with [math]q\in\mathbb T[/math], which appear as Di\c t\u a deformations of [math]W_4=F_2\otimes F_2[/math].

Apr 20'25

Given an Hadamard matrix [math]H\in M_5(\mathbb T)[/math], chosen dephased,

[[math]] H=\begin{pmatrix} 1&1&1&1&1\\ 1&a&x&*&*\\ 1&y&b&*&*\\ 1&*&*&*&*\\ 1&*&*&*&* \end{pmatrix} [[/math]]

prove that the numbers [math]a,b,x,y[/math] must satisfy the following equation:

[[math]] (x-y)(x-ab)(y-ab)=0 [[/math]]

Apr 20'25

Prove that the only Hadamard matrix at [math]N=5[/math] is the Fourier matrix,

[[math]] F_5=\begin{pmatrix} 1&1&1&1&1\\ 1&w&w^2&w^3&w^4\\ 1&w^2&w^4&w&w^3\\ 1&w^3&w&w^4&w^2\\ 1&w^4&w^3&w^2&w \end{pmatrix} [[/math]]

with [math]w=e^{2\pi i/5}[/math], up to the standard equivalence relation for such matrices.