# How to put this matrix in Jordan Canonical Form

Suppose you have the matrix $A = \begin{bmatrix} 1 & 0 & 1 & -1 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \\ \end{bmatrix}$

Put it into Jordan Canonical Form.

I get that the eigenvalue is 1 with multiplicity 4, the corresponding eigenvectors are $x = \begin{bmatrix}1\\0\\0\\0\end{bmatrix}$ and $y = \begin{bmatrix}0\\1\\0\\0\end{bmatrix}$.

When I try to construct the Jordan Block, however, I cannot figure out if the Jordan matrix has to consist of two 2x2 blocks or a 3x3 and 1x1 block. Because $(A-I)w = x$ and $(A-I)w=y$ are both inconsistent (for w$\in \mathbb{R^4}$). So by the theory we had so far about the Jordan Canonical Form, there can be no 2x2 blocks, let alone 3x3 blocks, for $(A - I)w$ is inconsistent for both x and y. How would you solve this problem?

Thanks!