I got characteristic polynomial $\displaystyle f(t)=(t^2)(t-\frac{\sqrt2}3)(t+\frac{\sqrt2}3)$

($\displaystyle \frac12 $, $\displaystyle \frac1{\sqrt2}$, $\displaystyle 1, \sqrt2$) is an eigenvector that is a basis for the eigenspace of $\displaystyle \lambda$= $\displaystyle \frac{\sqrt2}3$

($\displaystyle \frac12 $, $\displaystyle \frac{-1}{\sqrt2}$, $\displaystyle 1, -\sqrt2$) is an eigenvector that is a basis for the eigenspace of $\displaystyle \lambda$= $\displaystyle \frac{-\sqrt2}3$

but when I tried to find an eigenvector, $\displaystyle v $, to form a basis for the eigenspace corresponding to $\displaystyle \lambda =0$ I only got v=(0, 0, 0, 0) which cannot be an eigenvector and furthermore cannot be in a basis.... did I do something wrong or am I missing something?