The geometric multiplicity

Question

By given this matrix:

\begin{pmatrix}0&a&0\\0&0&1\\0&0&0\end{pmatrix}

Why for any a which is not 0 the geometric multiplicity = 1? and why for a = 0 the g.m. = 2?

I don't get it, and I'd like a short explanation how to calculate the geometric multiplicity in this case.

Thanks a lot

Answer 1

If the geometric multiplicity of $0$ is two which means that the dimension of the eigenspace of $0$ is $2$ then there's two linearly independent eigenvectors associated to $0$ and then the given matrix would be similar to $$\begin{pmatrix}0&0&0\\0&0&1\\0&0&0\end{pmatrix}$$ and this is a contradiction. Think to the rank of the two matrices!

Edit: Notice that two similar matrices have the same rank, dimension of the kernel, characteristic polynomial, minimal polynomial, trace...