# Locally free sheaf

A **complex Hadamard matrix** is any complex
matrix satisfying two conditions:

where denotes the Hermitian transpose of *H* and is the identity matrix. The concept is a generalization of the Hadamard matrix. 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 .

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 *N* (compare the real case, in which existence is not known for every *N*). For instance the Fourier matrices

belong to this class.

## 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,

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.

## References

- 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 links

- 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