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\pagestyle{fancy} \lhead{{\sf Homework 8 $|$ Due March 13
    (Friday) }} \rhead{\thepage} \cfoot{{\sf Math 8530 $|$ Linear
    Algebra $|$ Spring 2015 $|$ M.~Macauley}}


\begin{document}

$\;$

\noindent\emph{Read}: Lax, Chapter 6, pages 69--76. \\

\begin{enumerate}


% Lax Exercise 6.4-5
\item Let $A$ be an $n\times n$ matrix over $\C$ with an eigenvalue
  $\lambda$ of index $m\geq 2$ and corrresponding eigenvector $v_1$.
  Let $v_2$ be a generalized eigenvector satisfying $(A-\lambda
  I)v_2=v_1$.
  \begin{enumerate}
  \item Prove that for any natural number $N$,
    \[
    A^Nv_2=\lambda^N v_2+N\lambda^{N-1}v_1\,.
    \]
  \item Prove that for any polynomial $q(t)\in\C[t]$,
    \[
    q(A)v_2=q(\lambda)v_2+q'(\lambda)v_1\,,
    \]
    where $q'(t)$ is the derivative of $q$. 
  \item Conjecture a formula for $q(A)v_m$, where
    $v_1,\dots,v_m$ are generalized eigenvectors of $A$ with
    $(A-\lambda I)v_k=v_{k-1}$ (and say $v_0=0$, for convenience).
  \end{enumerate}



\item Consider the following matrices:
  \[
  A=\begin{bmatrix}-2&0\\0&-2\end{bmatrix}\,, \qquad\quad
  B=\begin{bmatrix}0&-2\\-2&0\end{bmatrix}\,, \qquad\quad
  C=\begin{bmatrix}3&1\\-1&1\end{bmatrix}\,.
  \]
  \begin{enumerate}
  \item Determine the characteristic and minimal polynomials of $A$,
    $B$, and $C$.
  \item  Determine the eigenvectors and generalized eigenvectors
    of $A$, $B$, and $C$.
  \end{enumerate}



\item Consider the following matrices:
  \[
  A=\begin{bmatrix} 2&-2&14\\0&3&-7\\0&0&2\end{bmatrix}\,,\qquad
  B=\begin{bmatrix} 0&-4&85\\1&4&-30\\0&0&3\end{bmatrix}\,,\qquad
  C=\begin{bmatrix} 2&2&1\\0&2&-1\\0&0&3\end{bmatrix}\,.
  \]
  A straightforward calculation shows that the characteristic polynomials are
  \[
  p_A(t)=p_B(t)=p_C(t)=(t-2)^2(t-3)\,.
  \]
  \begin{enumerate}
  \item Determine the minimal polynomials $m_A(t)$, $m_B(t)$, and
    $m_C(t)$.
  \item Determine the eigenvectors and generalized eigenvectors
    of $A$, $B$, and $C$.
  \item Determine which of these matrices are similar.
  \end{enumerate}



% Lax Lemma A15.2
\item Let $\lambda$ be an eigenvalue of $A$, and let $N_i$ be the
  nullspace of $(A-\lambda I)^i$. Prove that $A-\lambda I$ extends to
  a well-defined map $N_{i+1}/N_i\longrightarrow N_i/N_{i-1}$, and
  that this mapping is 1--1.



\item Let $A$ be an $n\times n$ matrix over $\C$.
  \begin{enumerate}
  \item Prove that if $A^k=A$ for some integer $k>1$, then $A$ is
    diagonalizable.
  \item Prove that if $A^k=0$, then $A^n=0$. \\
  \end{enumerate}

  
  
\item Let $X$ be an $n$-dimensional vector space over $\C$, and let
  $A,B\colon X\to X$ be linear maps.
  \begin{enumerate}
  \item Prove that if $AB=BA$, then for any eigenvector $v$ of
    $A$ with eigenvalue $\lambda$, the vector $Bv$ is an eigenvector
    of $A$ for $\lambda$.
  \item Show that if $\{A_1,A_2,\ldots\mid A_i\colon X\to X\}$ is
    a (possibly infinite) set of pairwise commuting maps, then there
    is a nonzero $x\in X$ that is an eigenvector of every $A_i$.
  \item Suppose that $A$ and $B$ are both diagonalizable. Show
    that $AB=BA$ if and only if they are \emph{simultaneously
      diagonalizable}, i.e., there exists an invertible $n\times
    n$-matrix $P$ such that both $P^{-1}AP$ and $P^{-1}BP$ are
    diagonal matrices.
  \end{enumerate}

\end{enumerate}

\end{document}







  
