one way to write x*y in matrix form is:
xHy (where xH is thus a row vector, instead of a column vector, and consists of the complex conjugates of coordinates of x)
thus (Mu)*(Mv) = (Mu)H(Mv) = uHMHMv.
if M is a complex orthogonal (unitary) matrix, then MHM = I,
whence (Mu)*(Mv) = uHv = u*v.
(your equation vH = v-1 makes no sense, vectors do not usually HAVE inverses, since vector multiplication is not generally defined).