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{\bf{Homework 9: Eigenvalues.}} 

\begin{enumerate}

\item Problem 9.1 from J\"anisch

\item Problem 9.2 from J\"anisch

\item Problem 9.3 from J\"anisch

\item Problem 9.1* from J\"anisch


\item Let $a_0, \dots , a_{n-1}$ be scalars.  Find the characteristic polynomial of the matrix 
$$\left | \begin{matrix}0  & 0 & 0 & \dots & 0 & -a_0\\
1 &0 &  0 & \dots & 0 & -a_1 \\
0 &1 &  0 & \dots & 0 & -a_2 \\
\dots & \dots & \dots & \dots &\dots  \\
0 & \dots &  0 & 1 &0 &  -a_{n-2}\\
0 & \dots &  & \dots  & 1 & -a_{n-1}) \end{matrix} \right |. $$
 
{\bf Hint. Look at the last problem of Homework 6.}

{\bf Remark.} This shows that any polynomial arises as the characteristic polynomial of some matrix (in case you were wondering). This matrix is called the \emph{Frobenius companion matrix} for a given polynomial.  

\item
\begin{enumerate}[(a)] 
\item Problem 10.1 from Janisch 
\item Find $A^5$ for the matrix $A$ from this problem. 
\end{enumerate}


\item 
\begin{enumerate}[(a)] 
\item Let $A:V\to V$ be a self-adjoint linear operator on a vector space $V$.
Prove that if $U$ is an invariant subspace for $A$, then $U^\perp$ is also invariant under $A$. 

\item Problem 10.3 from J\"anisch
\end{enumerate}

\item Let $J_\lambda$ be the $3\times 3$ Jordan block with the eigenvalue $\lambda$: 
$$J_\lambda= \left[\begin{matrix} \lambda & 1 & 0 \\ 0& \lambda & 1 \\ 0& 0&\lambda\end{matrix}\right].$$
Find $J_\lambda^2$, $J_\lambda^3$ and $J_\lambda^4$. 

{\bf Hint: It might help to use the binomial formula for matrices.}






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