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Math Help - Positive definiteness of an inverse

  1. #1
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    Positive definiteness of an inverse

    I would appreciate it if someone could just verify whether this works or not.

    Prove that if T is positive definite, then T^{-1} is positive definite.

    Since \langle T(x),x \rangle > 0 and T(T^{-1}(x))=x, we have 0 < \langle T(x),x \rangle = \langle T(x),T(T^{-1}(x)) \rangle = \langle T^{-1}(T(x)),T^{-1}(T(T^{-1}(x))) \rangle
    = \langle x,T^{-1}(x) \rangle = \langle T^{-1}(x),x \rangle. The last equality is true since \langle T(x),x \rangle is real, and also proves that T^{-1} is self-adjoint, and thus is positive definite.

    Thanks.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Re: Positive definiteness of an inverse

    Quote Originally Posted by AlexP View Post
    I would appreciate it if someone could just verify whether this works or not.

    Prove that if T is positive definite, then T^{-1} is positive definite.

    Since \langle T(x),x \rangle > 0 and T(T^{-1}(x))=x, we have 0 < \langle T(x),x \rangle = \langle T(x),T(T^{-1}(x)) \rangle = \langle T^{-1}(T(x)),T^{-1}(T(T^{-1}(x))) \rangle
    = \langle x,T^{-1}(x) \rangle = \langle T^{-1}(x),x \rangle. The last equality is true since \langle T(x),x \rangle is real, and also proves that T^{-1} is self-adjoint, and thus is positive definite.

    Thanks.
    This looks fine, but it may have been easier to just note that being positive definite is equivalent to having strictly positive eigenvalues, and since the eigenvalues of T^{-1} are the reciprocals of the eigenvalues of T everything quickly follows.
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  3. #3
    Super Member ILikeSerena's Avatar
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    Re: Positive definiteness of an inverse

    [QUOTE=AlexP;705251] \langle T(x),T(T^{-1}(x)) \rangle = \langle T^{-1}(T(x)),T^{-1}(T(T^{-1}(x))) \rangle

    I'm afraid this is not generally true.
    It requires T^{-1} to preserve the inner product, which is not automatically the case.
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