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\pagestyle{fancy} \lhead{{\sf Homework 7 $|$ Due October 5
    (Wednesday) }} \rhead{\thepage} \cfoot{{\sf Math 8530 $|$ Linear
    Algebra $|$ Fall 2016 $|$ M.~Macauley}}


\begin{document}

$\;$

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

\begin{enumerate}


\item Do the following for the matrix $A$ below, and then repeat it for $B$:
  \[
  A=\begin{bmatrix}-1&0&1&0\\2&1&2&1\\0&0&-1&0\\4&0&-6&1\end{bmatrix}
  \,,\qquad\qquad
  B=\begin{bmatrix}1&0&0&1\\2&1&0&-4\\1&0&1&-2\\0&0&0&1\end{bmatrix}\,.
  \] 
  \begin{enumerate}
    \item Find the characteristic polynomial and all (genuine) eigenvectors.
    \item For each eigenvalue $\lambda$, compute $\dim N_{(A-\lambda
      I)^j}$ for $j=1,2,3,\dots$.
    \item Find a basis of $\C^4$ consisting of generalized eigenvectors.
    \item Factor the matrix as a product $PJP^{-1}$, where the columns
      of $P$ are the generalized eigenvectors and $J$ is the matrix
      of the linear map with respect to this new basis. \bigskip
  \end{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). \bigskip
  \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. \bigskip



% Lax Exercise 6.10
\item Let $X$ be an $n$-dimensional vector space, and $A\colon X\to X$
  a linear map with distinct eigenvalues
  $\lambda_1,\dots,\lambda_n$. Let $v_1,\dots,v_n$ be the
  corresponding eigenvectors of $A$, and let $\ell_1,\dots,\ell_n$ be
  the corresponding eigenvectors of the transpose $A'\colon X'\to X'$.
  \begin{enumerate}
  \item Prove that $(\ell_i,v_i)\neq 0$ for $i=1,\dots,n$. 
  \item Show that if $x=a_1v_1+\cdots+a_nv_n$, then
    $a_i=(\ell_i,x)/(\ell_i,v_i)$.
  \item Is $\ell_1,\dots,\ell_n$ necessarily the dual basis of
    $v_1,\dots,v_n$? Why or why not? \medskip
  \end{enumerate}


\end{enumerate}

\end{document}







  
