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\begin{document}
\centerline{Math 201 Homework for Friday, Week 10}
\bigskip
Due: Friday, November 12.

\begin{prob}
  For each of the following matrices $A\in M_{n\times n}(F)$
  \begin{enumerate}[itemsep=2pt]
    \item[(i)] Determine all eigenvalues of $A$.
    \item[(ii)] For each eigenvalue $\lambda$ of $A$, find the set of eigenvectors corresponding to $\lambda$.
    \item[(iii)] If possible, find a basis for $F^n$ consisting of eigenvectors of $A$.
    \item[(iv)] If successful in finding such a basis, determine an invertible matrix $P$ and a diagonal matrix $D$ such that $A=PDP^{-1}$.
   \end{enumerate}
   \begin{enumerate}
    \item $A=\begin{pmatrix} 1 & 2 \\ 3 & 2\end{pmatrix}$ for $F = \R$.
    \item $A=\begin{pmatrix} 0 & -2 & -3\\ -1 & 1 & -1\\ 2 & 2 & 5\end{pmatrix}$ for $F = \R$.
    \item $A=\begin{pmatrix} 7 & -5\\ 10 & -7\end{pmatrix}$ for $F = \R$.
    \item $A=\begin{pmatrix} 7 & -5\\ 10 & -7\end{pmatrix}$ for $F = \C$.
    \item $A=\begin{pmatrix} 2 & 0 & -1\\ 4 & 1 & -4\\ 2& 0 & -1\end{pmatrix}$ for $F = \R$.
   \end{enumerate}
\end{prob}
\begin{prob}
    Let $f\colon V \to V$ be a linear transformation. For a positive integer
    $m$, we define $f^m$ inductively as $f\circ f^{m-1}$. Prove that if
    $\lambda$ is an eigenvalue for $f$, then $\lambda^m$ is an eigenvalue for
    $f^m$.
\end{prob}
\begin{prob}
    Let $T\colon M_{n\times n}(\R) \to M_{n\times n}(\R)$ defined as
    $T(A)=A^t$ (taking the transpose). One can prove that $T$ is a linear
    transformation.
   \begin{enumerate}
    \item Show that the only eigenvalues of $T$ are 1 and -1. (\emph{Hint:} Problem 2 might help.)
    \item For $n=2$, describe the eigenvectors corresponding to each eigenvalue. 
    \item Find an ordered basis $\mathcal{B}$ for $M_{2\times 2}(\R)$ such that the matrix that represents $T$ with respect to $\mathcal{B}$ is diagonal.
   \end{enumerate}
\end{prob}
\end{document}
