Let $r > 0$ be such that there exists $u \in \mathrm { SL } \left( \mathbb { R } ^ { 2 m } \right)$ satisfying $u \left( B ^ { 2 m } ( 1 ) \right) \subset B ^ { 2 m } ( r )$. Let $U \in \mathcal { M } _ { 2 m } ( \mathbb { R } )$ denote the matrix of $u$ in the canonical basis of $\mathbb { R } ^ { 2 m }$. Let $\lambda \in \mathbb { C }$ be a complex eigenvalue of the matrix $U$. Show that $| \lambda | \leqslant r$. For the case $\lambda$ non-real, if $P$ and $Q$ in $\mathcal { M } _ { 2 m , 1 } ( \mathbb { R } )$ are such that $Z = P + \mathrm { i } Q$ is an eigenvector column of $U$ for the eigenvalue $\lambda$, one may show that $\| U P \| ^ { 2 } + \| U Q \| ^ { 2 } = | \lambda | ^ { 2 } \left( \| P \| ^ { 2 } + \| Q \| ^ { 2 } \right)$.
Let $r > 0$ be such that there exists $u \in \mathrm { SL } \left( \mathbb { R } ^ { 2 m } \right)$ satisfying $u \left( B ^ { 2 m } ( 1 ) \right) \subset B ^ { 2 m } ( r )$. Let $U \in \mathcal { M } _ { 2 m } ( \mathbb { R } )$ denote the matrix of $u$ in the canonical basis of $\mathbb { R } ^ { 2 m }$. Let $\lambda \in \mathbb { C }$ be a complex eigenvalue of the matrix $U$. Show that $| \lambda | \leqslant r$.
For the case $\lambda$ non-real, if $P$ and $Q$ in $\mathcal { M } _ { 2 m , 1 } ( \mathbb { R } )$ are such that $Z = P + \mathrm { i } Q$ is an eigenvector column of $U$ for the eigenvalue $\lambda$, one may show that $\| U P \| ^ { 2 } + \| U Q \| ^ { 2 } = | \lambda | ^ { 2 } \left( \| P \| ^ { 2 } + \| Q \| ^ { 2 } \right)$.