Solution
= Solution
Choose the least $k\geq1$ such that
$$
(f-\alpha I)^kv=0.
$$
Then
$$
w=(f-\alpha I)^{k-1}v
$$
is nonzero by minimality, and
$$
(f-\alpha I)w=0.
$$
\b[Therefore $fw=\alpha w$, so $w$ is the required nonzero <eigenvector>.]
= Solution
Choose the least $k\geq1$ such that
$$
(f-\alpha I)^kv=0.
$$
Then
$$
w=(f-\alpha I)^{k-1}v
$$
is nonzero by minimality, and
$$
(f-\alpha I)w=0.
$$
\b[Therefore $fw=\alpha w$, so $w$ is the required nonzero <eigenvector>.]