- Butson-type Hadamard matrix
In mathematics, a complex
Hadamard matrix "H" of size "N" with all its columns (rows) mutuallyorthogonal , 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 m for quad} j,k=1,2,dots,N.
Existence
If "p" is prime then H(p,N) can existonly for N = mp with integer "m" andit is conjectured they exist for all such cases with p ge 3. In general, the problem of finding all setsq,N } such that the Butson - type matricesH(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 o infty one can approximate all
complex Hadamard matrices .*Fourier matrices F_N] _{jk}:= exp [(2pi i(j - 1)(k - 1) / N] {quad m for quad} 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_Notimes F_N in H(N,N^3).
:: D_{6} := egin{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 & -1 &-i & i& -1 & i \ 1 & i &-i & -i& i & -1 \ end{bmatrix}in H(4,6)
:: S_{6} := egin{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(2pi 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, 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=0200 Complex Hadamard Matrices of Butson type - a catalogue] , by Wojciech Bruzda, Wojciech Tadej and Karol Życzkowski, retrieved
October 24 ,2006
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