Suppose we have a matrix M, and some subspace W which is invariant under M. Suppose W = span(a, b) where a and b are vectors. If we know what the action M on a and b results in (say some linear combinations of a and b), how can we find eigenvectors of M using this information?
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I call $T$ the operator defined by $M$, which I take to be the matrix of $T$ in the canonical basis.
If $a$ and $b$ are linearly dependent, you're done: $W$ is one-dimensional and invariant: it is made of eigenvectors for $T$, hence for $M$, when written in the canonical basis.
If they are linearly independent, express $Aa$ and $Ab$ in terms of $a$ and $b$. You get a $2\times 2$ matrix, which represents the restriction of $T$ to $W$. Find eigenvectors for this $2\times 2$. Their coordinates are with respect to $(a,b)$. Go back to the canonical basis. You get eigenvectors of $M$.