equivalence relation
Two (complex) Hadamard
matrices are called equivalent, written
$H_1\simeq H_2$, if there
exist diagonal unitary matrices
$D_1$, $D_2$ and permutation matrices
$P_1$, $P_2$ such that:
$$H_1 = D_1 P_1 H_2 P_2 D_2.$$
Equivalence relation can be also defined with help of two monomial
(having exaclty one non-zero element in each column and each row)
[32]
unitary
matrices $$H_1= M_1 H_2 M_2.$$
Examples:
- All $2\times 2$ complex Hadamard matrices, the set of which reads
$$\Bigg\{
\left[\begin{array}{ll}e^{i\alpha_1}&0\\0&e^{i\alpha_2}\end{array}\right]
\cdot
\left[\begin{array}{rr}1&1\\1&-1\end{array}\right]
\cdot
\left[\begin{array}{ll}1&0\\0&e^{i\beta_2}\end{array}\right]\Bigg\}:\alpha_{1,2},\beta_2\in[0,2\pi)
$$
are equivalent to the Fourier
matrix $F_2$.
- All $3\times 3$ complex Hadamard matrices, the set of
which reads
$$\begin{aligned}
&\Bigg\{
\left[\begin{array}{lll}
e^{i\alpha_1}&0&0\\
0&e^{i\alpha_2}&0\\
0&0&e^{i\alpha_3}
\end{array}\right]
\cdot
\left[\begin{array}{lll}
1&1&1\\
1&w&w^2\\
1&w^2&w
\end{array}\right]
\cdot
\left[\begin{array}{lll}
1&0&0\\
0&e^{i\beta_2}&0\\
0&0&e^{1\beta_3}
\end{array}\right]
\Bigg\}\\
\bigcup \ &\Bigg\{
\left[\begin{array}{lll}
e^{i\alpha_1}&0&0\\
0&e^{i\alpha_2}&0\\
0&0&e^{i\alpha_3}
\end{array}\right]
\cdot
\left[\begin{array}{lll}
1&1&1\\
1&w^2&w\\
1&w&w^2
\end{array}\right]
\cdot
\left[\begin{array}{lll}
1&0&0\\
0&e^{i\beta_2}&0\\
0&0&e^{i\beta_3}
\end{array}\right]
\Bigg\}\\
&:\alpha_{1,2,3},\beta_{2,3}\in[0,2\pi), w=\exp(2\pi i/3)
\end{aligned}$$
are equivalent to the Fourier
matrix $F_3$.
See also: Haagerup-set, matrix fingerprint and
rectangular rank profile.