I recently found a proof of the spectral theorem for symmetric matrices that I had not seen before here. I am having difficulty understanding their proof of what they call "Lucky Fact 2" which shows that for a symmetric matrix the algebraic multiplicity of an eigenvalue is equal to its geometric multiplicity. I follow the proof of this until the very last line wherein the author writes.
...which should have been listed when we found the $v_{i}$
Which is meant to imply that $\begin{pmatrix}0 \\ 0 \\ \mathbf{u}\end{pmatrix}$ could not be an eigenvector of $A$ however I do not see why this is the case as $\begin{pmatrix}0 \\ 0 \\ \mathbf{u}\end{pmatrix}$ could simply equal $v_{1}$ or $v_{2}$ and be an eigenvector of $A$ with no issue.
Could someone explain to me why this is the case?