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

Printable View

- Nov 9th 2012, 10:57 AMsfspitfire23Hermitian 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 - Nov 9th 2012, 02:58 PMDevenoRe: Hermitian Matrix
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).