A **complex Hadamard matrix** is any complex
$N\times N$ matrix $H$ satisfying two conditions:

- unimodularity (the modulus of each entry is unity): $|H_{jk}|=1{\text{ for ))j,k=1,2,\dots ,N$
- orthogonality: $HH^{\dagger }=NI$,

where $\dagger$ denotes the Hermitian transpose of $H$ and $I$ is the identity matrix. The concept is a generalization of Hadamard matrices. Note that any complex Hadamard matrix $H$ can be made into a unitary matrix by multiplying it by ${\frac {1}{\sqrt {N))))$; conversely, any unitary matrix whose entries all have modulus ${\frac {1}{\sqrt {N))))$ becomes a complex Hadamard upon multiplication by ${\sqrt {N)).$

Complex Hadamard matrices arise in the study of operator algebras and the theory of quantum computation. Real Hadamard matrices and Butson-type Hadamard matrices form particular cases of complex Hadamard matrices.

Complex Hadamard matrices exist for any natural number $N$ (compare with the real case, in which Hadamard matrices do not exist for every $N$ and existence is not known for every permissible $N$). For instance the Fourier matrices (the complex conjugate of the DFT matrices without the normalizing factor),

- $[F_{N}]_{jk}:=\exp[2\pi i(j-1)(k-1)/N]{\quad {\rm {for\quad ))}j,k=1,2,\dots ,N$

belong to this class.

##
Equivalency

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

- $H_{1}=D_{1}P_{1}H_{2}P_{2}D_{2}.$

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 $N=2,3$ and $5$ all complex Hadamard matrices are equivalent to the Fourier matrix $F_{N))$. For $N=4$ there exists
a continuous, one-parameter family of inequivalent complex Hadamard matrices,

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

For $N=6$ the following families of complex Hadamard matrices
are known:

- a single two-parameter family which includes $F_{6))$,
- a single one-parameter family $D_{6}(t)$,
- a one-parameter orbit $B_{6}(\theta )$, including the circulant Hadamard matrix $C_{6))$,
- a two-parameter orbit including the previous two examples $X_{6}(\alpha )$,
- a one-parameter orbit $M_{6}(x)$ of symmetric matrices,
- a two-parameter orbit including the previous example $K_{6}(x,y)$,
- a three-parameter orbit including all the previous examples $K_{6}(x,y,z)$,
- a further construction with four degrees of freedom, $G_{6))$, yielding other examples than $K_{6}(x,y,z)$,
- a single point - one of the Butson-type Hadamard matrices, $S_{6}\in H(3,6)$.

It is not known, however, if this list is complete, but it is conjectured that $K_{6}(x,y,z),G_{6},S_{6))$ is an exhaustive (but not necessarily irredundant) list of all complex Hadamard matrices of order 6.