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Math Help - Spectral Decomposition and Polynomials

  1. #1
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    Spectral Decomposition and Polynomials

    Let T be a normal operator on a finite-dimensional complex inner product space V. Use the spectral decomposition to show that If g is a polynomial, then
    g(T) = g(m_1)T_1 + g(m_2)T_2 + ... + g(m_k)T_k, where the m_i's are the eigenvalues of T and the T_i's are the orthogonal projection of T on the eigenspace, W_i corresponding to the eigenvalue m_i.


    I know that from the spectral decomposition, we have T = (m_1)(T_1) + (m_2)(T_2) + ... + (m_k)(T_k). Hence, g(T) = g((m_1)(T_1)) + g((m_2)(T_2)) + ... + g((m_k)(T_k)). But how do i pull out the T_i's to obtain the above result?

    Thank You.
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  2. #2
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    Re: Spectral Decomposition and Polynomials

    Quote Originally Posted by H12504106 View Post
    Let T be a normal operator on a finite-dimensional complex inner product space V. Use the spectral decomposition to show that If g is a polynomial, then
    g(T) = g(m_1)T_1 + g(m_2)T_2 + ... + g(m_k)T_k, where the m_i's are the eigenvalues of T and the T_i's are the orthogonal projection of T on the eigenspace, W_i corresponding to the eigenvalue m_i.


    I know that from the spectral decomposition, we have T = (m_1)(T_1) + (m_2)(T_2) + ... + (m_k)(T_k).
    Start by computing monomials (powers of T):

    T^2 = (m_1T_1 + m_2T_2+\ldots+m_kT_k)(m_1T_1 + m_2T_2+\ldots+m_kT_k).

    Multiply out the brackets, remembering that the T_i are pairwise orthogonal projections so that  T_i^2=T_i and T_iT_j=0 if j\ne i. Thus

    T^2 = m_1^2T_1^2 + m_2^2T_2^2 + \ldots+m_k^2T_k^2.

    Use induction to prove the analogous result for each power of T. Finally, use linearity to deduce the result for a polynomial.
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