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Math Help - Transpose, orthogonal, and inverse

  1. #1
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    Transpose, orthogonal, and inverse

    Let u be an unit vector in R^{n} and H=I-2*u*u^{T}. Show that H is orthogonal, symmetric, and its own inverse.

    Assume H is orthogonal then H^{-1}=H^{T} and HH^{T}=I

    [(I-2*u*u^{T})(I-2*u*u^{T})^{T}]^{T}

    (I-2*u*u^{T})(I-2*u*u^{T})^{T}

    Now we have H*H^{T}=I

    I don't think everything is legal here.
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  2. #2
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    Symmetry:
    H^{T}=(I-2uu^{T})^{T}=I^{T}-2[u^{T}]^{T}u^{T}=I-2uu^{T}=H

    Inverse:
    H^{2}=(I-2uu^{T})*(I-2uu^{T})=I^{2}-4uu^{T}+4uu^{T}uu^{T}
    =I-4uu^{T}+4uu^{T}=I=HH^{-1}
    remember u is a unit vector.

    Orthogonal: Is H^{T}=H^{-1}?
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