# Tagged: Lie algebra

## 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 $F^N\mathbf{v}=\mathbf{0}$.

## Problem 68

Let $H$ and $E$ be $n \times n$ matrices satisfying the relation
$HE-EH=2E.$ Let $\lambda$ be an eigenvalue of the matrix $H$ such that the real part of $\lambda$ is the largest among the eigenvalues of $H$.
Let $\mathbf{x}$ be an eigenvector corresponding to $\lambda$. Then prove that
$E\mathbf{x}=\mathbf{0}.$