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A square matrix  $  H = \| h _ {ij} \| $
of order  $  n $, 
with entries  $  + 1 $
or  $  - 1 $, 
such that the equation

$$ \tag{* }
HH  ^ {T}  =  n I _ {n}  $$

holds, where  $  H  ^ {T} $
is the transposed matrix of  $  H $
and  $  I _ {n} $
is the unit matrix of order  $  n $. 
Equality (*) is equivalent to saying that any two rows of  $  H $
are orthogonal. Hadamard matrices have been named after J. Hadamard who showed [[#References|[1]]] that the determinant  $  | A | $
of a matrix  $  A = \| a _ {ij} \| $
of order  $  n $, 
with complex entries, satisfies the Hadamard inequality

$$ 
| A |  ^ {2}  \leq   \prod _ { i= } 1 ^ { n }  s _ {ii} ,
$$

where

$$ 
s _ {ik}  =  \sum _ {j = 1 } ^ { n } 
a _ {ij} \overline{ {a _ {kj} }}\; ,
$$

and  $  \overline{ {a _ {kj} }}\; $
is the element conjugate to  $  a _ {kj} $(
cf. [[Hadamard theorem|Hadamard theorem]] on determinants). In particular, if  $  | a _ {ij} | \leq  M $, 
then  $  | A | \leq  M  ^ {n} n  ^ {n/2} $. 
It follows that a Hadamard matrix is a square matrix consisting of  $  \pm  1 $'
s with maximal absolute value  $  n  ^ {n/2} $
of the determinant. The properties of Hadamard matrices are: 1)  $  HH  ^ {T} = nI _ {n} $
implies  $  H  ^ {T} H = nI _ {n} $
and vice versa; 2) transposition of rows or columns and multiplication of the elements of an arbitrary row or column by  $  - 1 $
again yields a Hadamard matrix; 3) the tensor product of two Hadamard matrices is also a Hadamard matrix, of order equal to the product of the orders of the factors. In other words, if  $  A = \| a _ {ij} \| $
and  $  B = \| b _ {ij} \| $
are Hadamard matrices of orders  $  m $
and  $  n $
respectively, then  $  C = \| a _ {ij} B \| $
is a Hadamard matrix of order  $  mn $. 
A Hadamard matrix with its first row and first column consisting only of  $  + 1 $
terms is said to be normalized. The order of a Hadamard matrix is  $  n = 1, 2 $
or  $  n \equiv 0 $(
$   \mathop{\rm mod}  4 $). 
The normalized Hadamard matrices of orders 1 and 2 are:

$$ 
[ 1],\ \ 
\left [

\begin{array}{cr}
 1  & 1  \\
 1  &- 1  \\
\end{array}

\right ] .
$$

The existence of a Hadamard matrix has been demonstrated for several classes of values of  $  n $(
see, for example, [[#References|[2]]], [[#References|[3]]]). At the time of writing (the 1980s), it has not yet been proved that a Hadamard matrix exists for any  $  n \equiv 0 $(
$   \mathop{\rm mod}  4 $). 
For methods of constructing Hadamard matrices see [[#References|[2]]]. Hadamard matrices are used in the construction of certain types of block designs [[#References|[2]]] and codes [[#References|[3]]] (cf. [[Block design|Block design]]; [[Code|Code]]). A Hadamard matrix of order  $  n = 4t $
is equivalent to a  $  ( 4t - 1, 2t - 1, t - 1 ) $-
design.

A generalized Hadamard matrix is a square matrix  $  H ( p, h) $
of order  $  h $, 
with as entries  $  p $-
th roots of unity, which satisfies the equality

$$ 
HH  ^ {cT}  =  h I _ {h} ,
$$

where  $  H  ^ {cT} $
is the conjugate transpose of the matrix  $  H $
and  $  I _ {h} $
is the unit matrix of order  $  h $. 
Generalized Hadamard matrices have properties analogous to 1) and 3) (cf. [[#References|[4]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Hadamard,   "Résolution d'une question relative aux déterminants"  ''Bull. Sci. Math. (2)'' , '''17'''  (1893)  pp. 240–246</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M. Hall,   "Combinatorial theory" , Blaisdell  (1967)  pp. Chapt. 14</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  W.W. Peterson,   "Error-correcting codes" , M.I.T. &amp; Wiley  (1961)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.T. Butson,   "Generalized Hadamard matrices"  ''Proc. Amer. Math. Soc.'' , '''13'''  (1962)  pp. 894–898</TD></TR></table>

====Comments====
Hadamard matrices are equivalent to so-called Hadamard  $  2 $-
designs; they are also important in statistical applications [[#References|[a6]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W.D. Wallis,   A.P. Street,   J.S. Wallis,   "Combinatorics: room squares, sum-free sets, Hadamard matrices" , Springer  (1972)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D.R. Hughes,   F.C. Piper,   "Design theory" , Cambridge Univ. Press  (1988)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  F.J. MacWilliams,   N.J.A. Sloane,   "The theory of error-correcting codes" , '''I-II''' , North-Holland  (1977)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  S.S. Agaian,   "Hadamard matrices and their applications" , ''Lect. notes in math.'' , '''1168''' , Springer  (1985)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  T. Beth,   D. Jungnickel,   H. Lenz,   "Design theory" , Cambridge Univ. Press  (1986)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  A. Hedayat,   W.D. Wallis,   "Hadamard matrices and their applications"  ''Ann. Stat.'' , '''6'''  (1978)  pp. 1184–1238</TD></TR></table>