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**Dinkydoe** So you understand how we found the relations:

$\displaystyle z= -2x, -3z = y$.

A vector that satisfies these relations is an eigenvector with $\displaystyle \lambda = -3$.

Now we may choose $\displaystyle x= c$ arbitrary. These relations give the values of y and z. We may choose x = c arbitrary because if v is an eigenvector, cv is an eigenvector as well.

Let x = c then by the above relations follows, $\displaystyle z=-2c, y = -6c$. Thus $\displaystyle v = (c,-2c,-6c) = c(1,-2,-6)$ is an eigenvector for any $\displaystyle c$.

The relations only show how $\displaystyle x,y,z$ relate to eachother. That's why one co-ordinate x,y or z may be freely chosen.