A complex Hadamard matrix is any complex N \times N matrix H satisfying two conditions: 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 ; conversely, any unitary matrix whose entries all have modulus 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), belong to this class.

Equivalency

Two complex Hadamard matrices are called equivalent, written, if there exist diagonal unitary matrices D_1, D_2 and permutation matrices P_1, P_2 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 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, For N=6 the following families of complex Hadamard matrices are known: 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.