one way to write x*y in matrix form is:

x^{H}y (where x^{H}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) = u^{H}M^{H}Mv.

if M is a complex orthogonal (unitary) matrix, then M^{H}M = I,

whence (Mu)*(Mv) = u^{H}v = u*v.

(your equation v^{H}= v^{-1}makes no sense, vectors do not usually HAVE inverses, since vector multiplication is not generally defined).