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Math Help - Eigenvectors of positive definite matrix

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    Eigenvectors of positive definite matrix

    Are the eigenvectors of positive definite matrix have all positive coordinates?
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    I know that eigenvalues of positive definite matrix are real positive. It does not say about eigenvectors. I
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    Oops, sorry. I misread your OP. I wouldn't think the components of an EV of a positive-definite matrix would have to be all positive. In fact, any non-zero scalar multiple of an eigenvector is an eigenvector. Hence, if all the components do happen to be positive, just multiply by a negative scalar, and you've got negative components.
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    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by krystoferivanov View Post
    Are the eigenvectors of positive definite matrix have all positive coordinates?
    Counterexample

    \begin{bmatrix}{1}&{0}\\{0}&{1}\end{bmatrix}\begin  {bmatrix}{-1}\\{-1}\end{bmatrix}=1\begin{bmatrix}{-1}\\{-1}\end{bmatrix}


    Fernando Revilla

    Edited: Sorry, I didn' see Ackbeet's post.
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