Matrices Satisfying $HF-FH=-2F$
Problem 69
Let $F$ and $H$ be an $n\times n$ matrices satisfying the relation
\[HF-FH=-2F.\]
(a) Find the trace of the matrix $F$.
(b) Let $\lambda$ be an eigenvalue of $H$ and let $\mathbf{v}$ be an eigenvector corresponding to $\lambda$. Show that there exists an positive integer $N$ such that $F^N\mathbf{v}=\mathbf{0}$.
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