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Math Help - An inner product space over C

  1. #1
    MHF Contributor Also sprach Zarathustra's Avatar
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    An inner product space over C

    Let V be an inner product space over field C(complex numbers). dim(V)<inf,
    T:V-->V linear transformation.

    1. Prove that T is normal (T*T=T*T) iff there exist polynomial p(x) in C[x] so that T*=p(T)

    2. Conclude that if T,S:V-->V linear transformations, T normal and TS=ST, so T*S=S*T

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  2. #2
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    Opalg's Avatar
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    Quote Originally Posted by Also sprach Zarathustra View Post
    Let V be an inner product space over field C(complex numbers). dim(V)<inf,
    T:V-->V linear transformation.

    1. Prove that T is normal (T*T=T*T) iff there exist polynomial p(x) in C[x] so that T*=p(T)
    If T is normal then it has a unitary diagonalisation. In other words, there is an orthonormal basis of V with respect to which T has a diagonal matrix (whose diagonal entries are the eigenvalues of T, but you don't need to know that for this question). Then the matrix of T* is also diagonal, and its diagonal entries are the complex conjugates of those of T. So you just need to show that there exists a polynomial taking a given finite set of complex numbers to their conjugates. If you don't know how to do that, do a Google search for "polynomial interpolation".

    Quote Originally Posted by Also sprach Zarathustra View Post
    2. Conclude that if T,S:V-->V linear transformations, T normal and TS=ST, so T*S=S*T
    This follows quite easily from 1. If S commutes with T, then it's easy to check that S commutes with any polynomial in T.
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  3. #3
    MHF Contributor Also sprach Zarathustra's Avatar
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    Thank you very much!
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