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I have this 4x4 matrix:

\begin{bmatrix} -1 & -1 & -1 & -1 \\ -1 & -1 & -1 & -1 \\ 4 & 3 & 4 & 2 \\ -2 & 0 & -2 & 1 \\ \end{bmatrix}

I know the eigenvectors are:

\begin{bmatrix} -1\\ -1 \\ 1 \\ 2 \\ \end{bmatrix} and \begin{bmatrix} -1\\ 0 \\ 1 \\ 0 \\ \end{bmatrix}

but I don't know how to get there. For matrices with patterns of upper right or lower left zeroes I've found 'easy' solutions, but this one doesn't seem to have an obvious way in.

To clarify a bit, I'm not just looking for specific steps for this matrix, but also general patterns that are useful to attack problems like this. For example, the solutions to this one are quite diverse, but involve row operations that are likely applicable only to that specific matrix.

Rodrigo de Azevedo
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MW Millar
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    you need to read about Jordan Normal Form because the characteristic polynomial and the minimal polynomial are both $(x-1)^3 ; x ; ; . ; ;$ Among Jordan Form problems this one is not at all bad – Will Jagy Jul 05 '20 at 22:38

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