Are the eigenvectors of positive definite matrix have all positive coordinates?

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- Jan 20th 2011, 07:05 AMkrystoferivanovEigenvectors of positive definite matrix
Are the eigenvectors of positive definite matrix have all positive coordinates?

- Jan 20th 2011, 07:12 AMAckbeet
- Jan 20th 2011, 07:24 AMkrystoferivanov
I know that eigenvalues of positive definite matrix are real positive. It does not say about eigenvectors. I

- Jan 20th 2011, 07:42 AMAckbeet
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.

- Jan 20th 2011, 07:43 AMFernandoRevilla
Counterexample

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

Fernando Revilla

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