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Thread: inner product

  1. #1
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    Wink inner product

    Please, help!
    Let V be a finite dimensional inner product space over C. Suppose that T is a positive operator on V ( called positive semidefinite by some authors ). Prove that T is invertible if and only if < Tv,v > > 0 for every v is in V \ { 0 }.
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  2. #2
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    How much do you know about positive operators? If you know that they are diagonalisable then this result follows fairly easily: There is an orthonormal basis of V with respect to which T has a diagonal matrix D, whose diagonal entries are the eigenvalues of T. The matrix D has an inverse if and only if none of these eigenvalues is 0. That is equivalent to the condition \langle Tv,v\rangle > 0 for all v≠0.
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