Prove that the matrix $A = \begin{pmatrix} 1&1&1 \\ -1&-1&-1 \\ 1&1&0 \end{pmatrix}$ is nilpotent, and find its invariants and Jordan form.
So far, I've verified directly that $A^3=0$, so $A$ is clearly nilpotent. We also know that the eigenvalues of any nilpotent matrix are $0$. Using the characteristic polynomial $\lambda^3=0$, I've confirmed this. We have an eigenvalue $0$ of algebraic multiplicity 3.
This is where I get stuck. I'm not sure how to find the invariants or Jordan form. I think you could take $J_i - \lambda_i I$ for all $i=1,2,3$, where $J_i$ are Jordan blocks, but I'm not sure how to find those. Any hints would be greatly appreciated.