Английская Википедия:Butson-type Hadamard matrix

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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\text{for}\quad j,k = 1,2,\dots,N.</math>

Existence

If p is prime and <math>N>1</math>, 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>. For <math>p=2</math>, the corresponding conjecture is existence for all multiples of 4. 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]

\text{ for }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, 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

• Complex Hadamard Matrices of Butson type - a catalogue, by Wojciech Bruzda, Wojciech Tadej and Karol Życzkowski, retrieved October 24, 2006

Шаблон:Matrix-stub