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Math Help - T is a projection and normal, then T is orthogonal projection

  1. #1
    Member Last_Singularity's Avatar
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    T is a projection and normal, then T is orthogonal projection

    Question: Let T be a normal operator on a finite-dimensional inner product space. Prove that if T is a projection, then T is also an orthogonal projection.

    Attempt: Since T is normal, TT^* = T^*T and since T is a projection, then T=T^2.

    Since T is an orthogonal projection if and only if T^2 = T = T^*, we already have the left equality by that fact that T is a projection. We only need to show that T=T^* and we'll be done.

    Somehow, I figure that I can do this by showing that <T(x),y>=<x,T(y)> but how?
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    Since T is normal, it is unitarily equivalent to a diagonal matrix. Since T=T^2, its eigenvalues can be only \pm 1. Thus T=UDU^* and T^*=U^{**}D^*U^*=UDU^*=T.
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