where denotes the Hermitian transpose of and is the identity matrix. The concept is a generalization of Hadamard matrices. Note that any complex Hadamard matrix 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
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,
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 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.
Referencesedit
U. Haagerup, Orthogonal maximal abelian *-subalgebras of the n×n matrices and cyclic n-roots, Operator Algebras and Quantum Field Theory (Rome), 1996 (Cambridge, MA: International Press) pp 296–322.
P. Dita, Some results on the parametrization of complex Hadamard matrices, J. Phys. A: Math. Gen. 37, 5355-5374 (2004).
F. Szollosi, A two-parametric family of complex Hadamard matrices of order 6 induced by hypocycloids, preprint, arXiv:0811.3930v2 [math.OA]
W. Tadej and K. Życzkowski, A concise guide to complex Hadamard matrices Open Systems & Infor. Dyn. 13 133-177 (2006)
External linksedit
For an explicit list of known complex Hadamard matrices and several examples of Hadamard matrices of size 7-16 see Catalogue of Complex Hadamard Matrices