real Hadamard matrix
A real Hadamard matrix is
any real $N\times N$ matrix $H$ that is unimodular
$|H_{jk}| \in\{-1,1\}$ and $HH^{\rm T} =N\mathbb{1}_N$
where $\mathbb{1}_N$ is the identity matrix of dimension $N$.
These matrices were considered by Sylvester in 1867 [98]
but their name is due to the 1893 work of Hadamard [63].
Quick facts on real Hadamard matrices.
- Necessary condition: a size $N$ of a real Hadamard matrix must satisfy
$N = 1$ or $N = 2$ or $N \equiv 0 \bmod 4$.
- The Hadamard conjecture states that there exists a real Hadamard
matrix of size $N$ for every $N \equiv 0 \bmod 4$.
- After the 2005 discovery
[73]
of a real Hadamard matrix
for $N = 428$
the case $N = 668$ is the smallest
size for which the existence problem remains open.
- Real Hadamard matrices obtained from a given matrix $H$
by permuting or multiplying by $-1$ any of its rows or columns are called
equivalent.
- For $N=2$, $4$, $8$ and $12$ all real Hadamard matrices
are equivalent.
- The number $E$ of equivalence classes
reads [32], [62]:
| $N =$ |
$1$ |
$2$ |
$4$ |
$8$ |
$12$ |
$16$ |
$20$ |
$24$ |
$28$ |
$32$ |
$36$ |
| $E =$ |
$1$ |
$1$ |
$1$ |
$1$ |
$1$ |
$5$ |
$3$ |
$60$ |
$487$ |
$13710027$ |
$4745357^*$ |
while the number decorated with a star provides lower bounds
only - see the work of Orrick [21].
More information on real Hadamard matrices: