On integer matrices obeying certain matrix equations

We discuss integer matrices B of odd order v which satisfy Br = ± B, BBr = vI - J, BJ = O. Matrices of this kind which have zero diagonal and other elements ± 1 give rise to skew-Hadamard and n-type matrices; we show that the existence of a skew-Hadamard (n-type) matrix of order h implies the existence of skew-Hadamard (n-type) matrices of orders (h - 1)5 + 1 and (h - 1)7 + 1. Finally we show that, although there are matrices B with elements other than ± 1 and 0, the equations force considerable restrictions on the elements of B.