Butson-type Hadamard matrix

In mathematics, a complex Hadamard matrix H of size N with all its columns (rows) mutually orthogonal, belongs to the Butson-type H(q, N) if all its elements are powers of q-th root of unity,

:: $(H_{jk})^q = 1 \quad\text{for}\quad j,k = 1,2,\dots,N.$

Existence

If p is prime and $N>1$ , then $H(p,N)$ can exist

only for $N = mp$ with integer m and

it is conjectured they exist for all such cases

with $p \ge 3$ . For $p=2$ , the corresponding conjecture is existence for all multiples of 4.

In general, the problem of finding all sets

$\{q,N \}$ such that the Butson-type matrices

$H(q,N)$ exist, remains open.

Examples

$H(2,N)$ contains real Hadamard matrices of size N,
$H(4,N)$ contains Hadamard matrices composed of $\pm 1, \pm i$ – such matrices were called by Turyn, complex Hadamard matrices.
in the limit $q \to \infty$ one can approximate all complex Hadamard matrices.
Fourier matrices $[F_N]_{jk}:= \exp[(2\pi i (j-1)(k-1)/N]$

\text{ for }j,k = 1,2,\dots,N

: belong to the Butson-type,

:: $F_N \in H(N,N),$

: while

:: $F_N \otimes F_N \in H(N,N^2),$

:: $F_N \otimes F_N\otimes F_N \in H(N,N^3).$

:: $D_6 := \begin{bmatrix}
1 & 1 & 1 & 1 & 1 & 1 \\
1 & -1 & i & -i& -i & i \\
1 & i &-1 & i& -i &-i \\
1 & -i & i & -1& i &-i \\
1 & -i &-i & i& -1 & i \\
1 & i &-i & -i& i & -1 \\
\end{bmatrix}
\in\, H(4,6)$ ,

:: $S_6 := \begin{bmatrix}
1 & 1 & 1 & 1 & 1 & 1 \\
1 & 1 & z & z & z^2 & z^2 \\
1 & z & 1 & z^2&z^2 & z \\
1 & z & z^2& 1& z & z^2 \\
1 & z^2& z^2& z& 1 & z \\
1 & z^2& z & z^2& z & 1 \\
\end{bmatrix}
\in\, H(3,6)$

: where $z =\exp(2\pi i/3).$

References

A. T. Butson, Generalized Hadamard matrices, Proc. Am. Math. Soc. 13, 894-898 (1962).
A. T. Butson, Relations among generalized Hadamard matrices, relative difference sets, and maximal length linear recurring sequences, Can. J. Math. 15, 42-48 (1963).
R. J. Turyn, Complex Hadamard matrices, pp. 435–437 in Combinatorial Structures and their Applications, Gordon and Breach, London (1970).

External links

[http://chaos.if.uj.edu.pl/~karol/hadamard/index.php?I=chm_butson Complex Hadamard Matrices of Butson type - a catalogue], by Wojciech Bruzda, Wojciech Tadej and Karol Życzkowski, retrieved October 24, 2006

Category:Matrices (mathematics)