Complex Hadamard Matrix

Equivalency

Two complex Hadamard matrices are called equivalent, written, if there exist diagonal unitary matrices and permutation matrices such that

Any complex Hadamard matrix is equivalent to a dephased Hadamard matrix, in which all elements in the first row and first column are equal to unity.

For and all complex Hadamard matrices are equivalent to the Fourier matrix . For there exists a continuous, one-parameter family of inequivalent complex Hadamard matrices,

$F_{4}^{(1)}(a):= \begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & ie^{ia} & -1 & -ie^{ia} \\ 1 & -1 & 1 &-1 \\ 1 & -ie^{ia}& -1 & i e^{ia} \end{bmatrix} {\quad \rm with \quad } a\in [0,\pi) .$

For the following families of complex Hadamard matrices are known:

a single two-parameter family which includes ,
a single one-parameter family ,
a one-parameter orbit, including the circulant Hadamard matrix ,
a two-parameter orbit including the previous two examples ,
a one-parameter orbit of symmetric matrices,
a two-parameter orbit including the previous example ,
a three-parameter orbit including all the previous examples ,
a further construction with four degrees of freedom, yielding other examples than ,
a single point - one of the Butson-type Hadamard matrices, .

It is not known, however, if this list is complete, but it is conjectured that is an exhaustive (but not necessarily irredundant) list of all complex Hadamard matrices of order 6.

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Complex Hadamard Matrix - Equivalency