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A square matrix of order , with entries or , such that the equation

 (*)

holds, where is the transposed matrix of and is the unit matrix of order . Equality (*) is equivalent to saying that any two rows of are orthogonal. Hadamard matrices have been named after J. Hadamard who showed [1] that the determinant of a matrix of order , with complex entries, satisfies the Hadamard inequality

where

The existence of a Hadamard matrix has been demonstrated for several classes of values of (see, for example, [2], [3]). At the time of writing (the 1980s), it has not yet been proved that a Hadamard matrix exists for any (). For methods of constructing Hadamard matrices see [2]. Hadamard matrices are used in the construction of certain types of block designs [2] and codes [3] (cf. Block design; Code). A Hadamard matrix of order is equivalent to a -design.

A generalized Hadamard matrix is a square matrix of order , with as entries -th roots of unity, which satisfies the equality

where is the conjugate transpose of the matrix and is the unit matrix of order . Generalized Hadamard matrices have properties analogous to 1) and 3) (cf. [4]).

#### References

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