About the consecutive dimensions of k-eigenspaces

Question

If $\lambda$ is an eigenvalue of matrix $A$, is it possible that for some positive integer $n$, dim($N(A-\lambda I)^{n+1})$-dim($N(A-\lambda I)^{n})>1$.
I am studying generalised eigenspaces and in all the examples I have read so far, dimension is increasing by 1 as power of $A-\lambda I$ is increasing until it hits the algebraic multiplicity of $\lambda$.

Answer 1

Consider $\lambda=0$, $n=1$ and $A=\begin{pmatrix} 0&1&0&0 \\ 0&0&0&0\\0&0&0&1\\0&0&0&0 \end{pmatrix}$