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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  that the determinant of a matrix of order , with complex entries, satisfies the Hadamard inequality where and is the element conjugate to (cf. Hadamard theorem on determinants). In particular, if , then . It follows that a Hadamard matrix is a square matrix consisting of 's with maximal absolute value of the determinant. The properties of Hadamard matrices are: 1) implies and vice versa; 2) transposition of rows or columns and multiplication of the elements of an arbitrary row or column by 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 and are Hadamard matrices of orders and respectively, then is a Hadamard matrix of order . A Hadamard matrix with its first row and first column consisting only of terms is said to be normalized. The order of a Hadamard matrix is or ( ). The normalized Hadamard matrices of orders 1 and 2 are: The existence of a Hadamard matrix has been demonstrated for several classes of values of (see, for example, , ). 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 . Hadamard matrices are used in the construction of certain types of block designs  and codes  (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. ).