A set \(T\) oxf cells of a Hadamard matrix \(H\) is called a trade if there is another Hadamard matrix \(H'\) that differs from \(H\) in \(T\) only; in particular, \(H\) is not uniquely reconstructed from its values out of \(T\) . In this paper, we show that a generalized Hadamard matrix has a diagonal trade if and only if it has a nice symmetric structure called consta-skew. In particular, all other generalized Hadamard matrices are uniquely reconstructed from the off-diagonal elements. A similar result is proved for complex Hadamard matrices, involving the concept of mixed-skew matrices. In addition to skew-type matrices (in the well-known sense), only few consta-skew generalized and mixed-skew complex Hadamard matrices are known. For generalized Hadamard matrices over \((Z_q,+)\) , we prove the nonexistence of circulant consta-skew matrices.
