# Locally free sheaf

A complex Hadamard matrix is any complex ${\displaystyle N\times N}$ matrix ${\displaystyle H}$ satisfying two conditions:

where ${\displaystyle {\dagger }}$ denotes the Hermitian transpose of H and ${\displaystyle {\mathbb {I} }}$ is the identity matrix. The concept is a generalization of the Hadamard matrix. Note that any complex Hadamard matrix ${\displaystyle H}$ can be made into a unitary 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 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

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

belong to this class.

## Equivalency

Two complex Hadamard matrices are called equivalent, written ${\displaystyle H_{1}\simeq H_{2}}$, if there exist diagonal unitary matrices ${\displaystyle D_{1},D_{2}}$ and permutation 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 a dephased 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 exists a continuous, one-parameter family of inequivalent complex Hadamard matrices,

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

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

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

## 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)