In mathematics, a complex Hadamard matrix

Hadamard matrix

In mathematics, an 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...

 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,

Existence

If p is prime

Prime number

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

 then can exist
only for with integer m and
it is conjectured they exist for all such cases
with .
In general, the problem of finding all sets
such that the Butson - type matrices
exist, remains open.

Examples

• contains real Hadamard matrices of size N,

• contains Hadamard matrices composed of - such matrices were called by Turyn, complex Hadamard matrices.

• in the limit one can approximate all complex Hadamard matrices.

• Fourier matrices
  belong to the Butson-type,
  

  

  
  

  while

  
  

  

  
  

  

  
  

  
  

  , where
  

  External links

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

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