
Butson-type Hadamard matrices
Encyclopedia
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,
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.
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 primePrime 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

only for

it is conjectured they exist for all such cases
with

In general, the problem of finding all sets


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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