\documentclass[12pt]{article}
\usepackage{fancyhdr,fancybox,amssymb,epsfig, amsmath, latexsym}
\usepackage[all]{xy}
\topmargin=-.5in
\headsep=0.2in
\oddsidemargin=0in
\textwidth=6.7in
\textheight=9.2in
\footskip=.5in

\newcommand{\C}{\mathbb{C}}
\newcommand{\F}{\mathbb{F}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Z}{\mathbb{Z}}
\DeclareMathOperator{\im}{Im}
\DeclareMathOperator{\Span}{span}
\DeclareMathOperator{\Char}{char}
\DeclareMathOperator{\rank}{rank}
\DeclareMathOperator{\sgn}{sgn}
\DeclareMathOperator{\tr}{tr}

\newcommand{\vv}[2]{\begin{bmatrix} #1 \\ #2 \end{bmatrix}}
\newcommand{\vvv}[3]{\begin{bmatrix} #1 \\ #2 \\ #3 \end{bmatrix}}
\newcommand{\vvvv}[4]{\begin{bmatrix} #1 \\ #2 \\ #3 \\ #4 \end{bmatrix}}

\renewcommand{\footrulewidth}{1pt}
\newenvironment{proof}[1][Proof.]{\medskip\begin{trivlist}
    \item[\hskip \labelsep {\sffamily #1}]}{\qed\end{trivlist}\bigskip}
\newenvironment{sol}[1][Solution.]{\medskip\begin{trivlist}
    \item[\hskip \labelsep {\sffamily #1}]}{\qed\end{trivlist}\bigskip}
\newenvironment{fminipage}%
{\setlength{\fboxsep}{15pt}\begin{Sbox}\begin{minipage}}%
{\end{minipage}\end{Sbox}\fbox{\TheSbox}}
\newcommand{\qed}{\hfill \ensuremath{\Box}}
\newcommand{\disp}{\displaystyle}
\newcommand{\ds}{\displaystyle}


\pagestyle{fancy} \lhead{{\sf Homework 7 $|$ Due October 13
    (Friday) }} \rhead{\thepage} \cfoot{{\sf Math 8530 $|$ Linear
    Algebra $|$ Fall 2023 $|$ M.~Macauley}}


\begin{document}

$\;$ \vspace{-2mm}

\noindent\emph{Read}: Lax, Chapter 6, pages 58--76. \medskip\vspace{0mm}

\begin{enumerate}

%%------------------------------------------------------------------------
  
\item Let $A\colon X\to X$ be a linear map with distinct eigenvalues
  $\lambda_1,\dots,\lambda_n$ and corresponding eigenvectors
  $v_1,\dots,v_n$. Let $\ell_1,\dots,\ell_n$ be the dual basis.
  \begin{enumerate}
  \item Prove that $\ell_1,\dots,\ell_n$ are eigenvectors of the
    transpose map $A'\colon X'\to X'$.
  \item Now, suppose that $f_1,\dots,f_n$ is \emph{any} basis of eigenvectors
    of $A'$. Prove that $(f_i,v_j)=0$ if $i\neq j$ and $(f_i,v_i)\neq 0$.
  \item For any $x=a_1v_1+\cdots+a_nv_n$, derive a formula for $a_i$ in
    terms of $x$, $v_i$, and $f_i$. \medskip
    \end{enumerate}

%%------------------------------------------------------------------------  
  
\item Let $A$ be an invertible $n\times n$ matrix. Prove that $A^{-1}$
  can be written as a polynomial in degree at most $n-1$. That is,
  prove that there are scalars $c_i$ such that
  \[
  A^{-1}=c_{n-1}A^{n-1}+c_{n-2}A^{n-2}+\cdots+c_1A+c_0I. 
  \] 
  %\medskip

%%------------------------------------------------------------------------ 

\item Let $\lambda$ be an eigenvalue of $A$, and let $N_j$ be the
  nullspace of $(A-\lambda I)^j$. Elements of $N_j$ are called
  \emph{generalized eigenvectors} of $\lambda$. The special case of
  $j=1$ are the ordinary (``genuine'') eigenvectors. Prove that
  $A-\lambda I$ extends to a well-defined map
  $N_{j+1}/N_j\longrightarrow N_j/N_{j-1}$, and that this mapping is
  1--1. \medskip

%%------------------------------------------------------------------------ 
 
\item Let $A$ be an $n\times n$ matrix over $\C$ with an eigenvalue
  $\lambda$ and corrresponding eigenvector $v_1$.  Let $v_2$ be a
  generalized eigenvector satisfying $(A-\lambda I)v_2=v_1$.
  \begin{enumerate}
  \item Show that $A^Nv_2=\lambda^N v_2+N\lambda^{N-1}v_1$, for any $N\in\N$. 
  \item Show that $q(A)v_2=q(\lambda)v_2+q'(\lambda)v_1$,
    for any polynomial $q(t)\in\C[t]$. 
  \item Give a formula (no proof needed) for $q(A)v_m$, where
    $v_1,\dots,v_m$ are generalized eigenvectors of $A$ with
    $(A-\lambda I)v_k=v_{k-1}$. Let $v_0=0$, for convenience. \medskip
  \end{enumerate}

%%------------------------------------------------------------------------ 

\item Do the following for the matrix $A$ below, and then repeat it for $B$:
  \[
  A=\begin{bmatrix}-1&0&1&0\\2&1&2&1\\0&0&-1&0\\4&0&-6&1\end{bmatrix}
  \,,\qquad\quad
  B=\begin{bmatrix}1&0&0&1\\2&1&0&-4\\1&0&1&-2\\0&0&0&1\end{bmatrix}
  \,,\qquad\quad
  J_\lambda=\begin{bmatrix}\lambda&1&&\\&\lambda&\ddots& \\ &&\ddots&1\\ &&&\lambda\end{bmatrix}\,.
  \] 
  \begin{enumerate}
    \item Find the characteristic and minimal polynomials, 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 $\mathcal{B}$ of $\C^4$ consisting of generalized
      eigenvectors, so that the matrix with respect to this basis is
      $J=P^{-1}AP$, where $J$ is a \emph{Jordan matrix}. This means
      that $J$ is block-diagonal formed from \emph{Jordan blocks}
      $J_\lambda$; see above.
    \item A subspace $Y\subseteq\C^4$ is $A$-\emph{invariant} if
      $A(Y)\subseteq Y$. Of the $16$ subspaces spanned by subsets of
      $\mathcal{B}$, determine which ones are $A$-invariant.
  \end{enumerate}
  
%%------------------------------------------------------------------------

\end{enumerate}

\end{document}


