Stiles–Crawford effect

2014-01-22T13:30:27Z

130.235.172.20: changed double negation + type

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,

:: <math>(H_{jk})^q=1 {\quad \rm for \quad} j,k=1,2,\dots,N. </math>

== Existence ==

If ''p'' is [[Prime number|prime]] then <math>H(p,N)</math> can exist
only for <math>N = mp</math> with integer ''m'' and
it is conjectured they exist for all such cases
with <math>p \ge 3</math>.
In general, the problem of finding all sets
<math>\{q,N \}</math> such that the Butson - type matrices
<math>H(q,N)</math> exist, remains open.

== Examples ==
*<math>H(2,N)</math> contains real [[Hadamard matrices]] of size ''N'',

*<math>H(4,N)</math> contains Hadamard matrices composed of <math>\pm 1, \pm i</math> - such matrices were called by Turyn, complex Hadamard matrices.

* in the limit <math>q \to \infty </math> one can approximate all [[complex Hadamard matrices]].

*'''Fourier''' matrices <math> [F_N]_{jk}:= \exp[(2\pi i(j - 1)(k - 1) / N]
{\quad \rm for \quad} j,k=1,2,\dots,N </math>
belong to the Butson-type,

:: <math>F_N \in H(N,N),</math>

: while

:: <math>F_N \otimes F_N \in H(N,N^2),</math>

:: <math>F_N \otimes F_N\otimes F_N \in H(N,N^3).</math>

:: <math> 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)
</math>

:: <math> 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)
</math>, where <math>z =\exp(2\pi i/3).</math>

==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, Canad. 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]]

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Stiles–Crawford effect