In mathematics, a complex

Hadamard matrix In mathematics, a Hadamard matrix, named after the French mathematician Jacques Hadamard, is a square matrix whose entries are either +1 or −1 and whose rows are mutually orthogonal. In geometric terms, this means that each pair of rows in ...

''H'' of size ''N'' with all its columns (rows) mutually

orthogonal In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...

, belongs to the Butson-type ''H''(''q'', ''N'') if all its elements are powers of ''q''-th root of unity, ::

(H_)^q=1  j,k=1,2,\dots,N.

Existence

If ''p'' is

prime A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

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

\

such that the Butson - type matrices

H(q,N)

exist, remains open.

Examples

H(2,N)

contains real

Hadamard matrices In mathematics, a Hadamard matrix, named after the French mathematician Jacques Hadamard, is a square matrix whose entries are either +1 or −1 and whose rows are mutually orthogonal. In geometric terms, this means that each pair of rows in ...

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 A complex Hadamard matrix is any complex N \times N matrix H satisfying two conditions: *unimodularity (the modulus of each entry is unity): , H_, =1 j,k=1,2,\dots,N *orthogonality: HH^ = NI , where denotes the Hermitian transpose of H and ...

. *Fourier matrices

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_ := 
\begin 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
\in H(4,6)

S_ := 
\begin 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

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