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Math Help - Spectral decomposition and invertibility

  1. #1
    Member Last_Singularity's Avatar
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    Spectral decomposition and invertibility

    Question: Let T be a normal operator on a finite-dimensional complex inner product space V. Consider spectral decomposition T = \lambda_1 T_1 + ... + \lambda_k T_k and prove that T is invertible if and only if \lambda_i \neq 0 for i \leq i \leq k.

    Any pointers would be helpful - I am unable to begin currently.
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    Quote Originally Posted by Last_Singularity View Post
    Question: Let T be a normal operator on a finite-dimensional complex inner product space V. Consider spectral decomposition T = \lambda_1 T_1 + ... + \lambda_k T_k and prove that T is invertible if and only if \lambda_i \neq 0 for i \leq i \leq k.

    Any pointers would be helpful - I am unable to begin currently.

    Well, this is ALWAYS true: any operator (of a finite-dimensional vector space V) over any field is invertible iff all its eigenvalues are different from zero, and the proof is painfully simple: T is NOT invertible iff Ker(T)\ne {0}\,\Longleftrightarrow\,\exists\,0\ne v\in V\,\,\,s.t.\,\,Tv=0=0\cdot v\,\Longleftrightarrow\,0 is an eigenvaue of T.

    Tonio
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