  The circulant real and complex matrices are used to find new real and complex
conference matrices. With them we construct Sylvester inverse orthogonal
matrices by doubling the size of inverse complex conference matrices. When the
free parameters take values on the unit circle the inverse orthogonal matrices
transform into complex Hadamard matrices. The method is used for $n=6$
conference matrices and in this way we find new parametrisations of Hadamard
matrices for dimension $ n=12$.
