# Eigenvectors of positive definite matrix

• Jan 20th 2011, 07:05 AM
krystoferivanov
Eigenvectors of positive definite matrix
Are the eigenvectors of positive definite matrix have all positive coordinates?
• Jan 20th 2011, 07:12 AM
Ackbeet
• Jan 20th 2011, 07:24 AM
krystoferivanov
I know that eigenvalues of positive definite matrix are real positive. It does not say about eigenvectors. I
• Jan 20th 2011, 07:42 AM
Ackbeet
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 AM
FernandoRevilla
Quote:

Originally Posted by krystoferivanov
Are the eigenvectors of positive definite matrix have all positive coordinates?

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.