Let M be an nxn complex orthogonal matrix. Show that for any two column vectors v, w in C,
show that v * w = (Mv) * (Mw)
where * denotes the dot product.
I'm stuck. I know that any vector v^H = v^-1
Thanks
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).