Thread: Hermitian Matrix

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^Hy (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^HM^HMv.

if M is a complex orthogonal (unitary) matrix, then M^HM = I,

whence (Mu)*(Mv) = u^Hv = u*v.

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