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Math Help - Hermitian Matrix

  1. #1
    Senior Member sfspitfire23's Avatar
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    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
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  2. #2
    MHF Contributor

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    Re: Hermitian Matrix

    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).
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