Acomplex Hadamard matrix is anycomplex${\displaystyle N\times N}$matrix${\displaystyle H}$ satisfying two conditions:

• unimodularity (themodulus of each entry is unity):${\displaystyle |H_{jk}|=1\text{ for }j,k=1,2,\dots ,N}$
• orthogonality:${\displaystyle H{H}^{†}=NI}$,

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

Complex Hadamard matrices arise in the study ofoperator algebras and the theory ofquantum computation.Real Hadamard matrices andButson-type Hadamard matrices form particular cases of complex Hadamard matrices.

Complex Hadamard matrices exist for anynatural number${\displaystyle N}$ (compare with the real case, in which Hadamard matrices do not exist for every${\displaystyle N}$ andexistence is not known for every permissible${\displaystyle N}$). For instance the Fourier matrices (thecomplex conjugate of theDFT matrices without the normalizing factor),

${\displaystyle [F_N{]}_{jk}:=\exp [2\pi i(j-1)(k-1)/N]\mathrm{f}\mathrm{o}\mathrm{r}j,k=1,2,\dots ,N}$

belong to this class.

Two complex Hadamard matrices are calledequivalent, written${\displaystyle H_1\simeq H_2}$, if there existdiagonal unitary matrices${\displaystyle D_1,D_2}$ andpermutation matrices${\displaystyle P_1,P_2}$ such that

${\displaystyle H_1=D_1{P}_1{H}_2{P}_2{D}_2.}$

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

For${\displaystyle N=2,3}$ and${\displaystyle 5}$ all complex Hadamard matrices are equivalent to the Fourier matrix${\displaystyle F_N}$. For${\displaystyle N=4}$ there existsa continuous, one-parameter family of inequivalent complex Hadamard matrices,

For${\displaystyle N=6}$ the following families of complex Hadamard matricesare known:

• a single two-parameter family which includes${\displaystyle F_6}$,
• a single one-parameter family${\displaystyle D_6(t)}$,
• a one-parameter orbit${\displaystyle B_6(\theta )}$, including thecirculant Hadamard matrix${\displaystyle C_6}$,
• a two-parameter orbit including the previous two examples${\displaystyle X_6(\alpha )}$,
• a one-parameter orbit${\displaystyle M_6(x)}$ ofsymmetric matrices,
• a two-parameter orbit including the previous example${\displaystyle K_6(x,y)}$,
• a three-parameter orbit including all the previous examples${\displaystyle K_6(x,y,z)}$,
• a further construction with four degrees of freedom,${\displaystyle G_6}$, yielding other examples than${\displaystyle K_6(x,y,z)}$,
• a single point - one of the Butson-type Hadamard matrices,${\displaystyle S_6\in H(3,6)}$.

It is not known, however, if this list is complete, but it isconjectured that${\displaystyle K_6(x,y,z),G_6,S_6}$ is an exhaustive (but not necessarily irredundant) list of all complex Hadamard matrices of order 6.

• For an explicit list of known${\displaystyle N=6}$ complex Hadamard matrices and several examples of Hadamard matrices of size 7-16 seeCatalogue of Complex Hadamard Matrices

• Matrices (mathematics)