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\pagestyle{fancy} \lhead{{\sf Homework 7 $|$ Due October 9
    (Friday) }} \rhead{\thepage} \cfoot{{\sf Math 8530 $|$ Linear
    Algebra $|$ Fall 2020 $|$ M.~Macauley}}


\begin{document}

$\;$

\noindent\emph{Read}: Lax, Chapter 6, pages 58--69. \medskip\vspace{1mm}

\begin{enumerate}

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

\item Let $A$ be an $n\times n$ matrix over $\C$ with distinct
  eigenvalues $\lambda_1,\dots,\lambda_n$. For a vector
  $z=(z_1,\dots,z_n)\in\C^n$, define the \emph{norm} of $z$ by
  \[
  ||z||=\Big(\sum_{i=1}^n|z_i|^2\Big)^{1/2}.
  \]
  \begin{enumerate}
  \item State and prove a sufficient condition for
    $\displaystyle\lim_{N\to\infty}\!||A^Nz||=0$ for all $z\in\C^n$.
  \item State and prove a sufficient condition for
    $\displaystyle\lim_{N\to\infty}\!||A^Nz||=\infty$ for all
    $z\in\C^n$.
  \end{enumerate}
  When we study inner product spaces, we will be able to state and
  prove necessary conditions as well. \bigskip

%%------------------------------------------------------------------------
  
\item Suppose that $B=PAP^{-1}$, and $A$ has eigenvalues
  $\lambda_1,\dots,\lambda_n$ and eigenvectors $v_1,\dots,v_n$. What
  are the eigenvalues and eigenvectors of $B$? Prove your
  claims. \bigskip

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

\item Let $X$ be an $n$-dimensional vector space, and $A\colon X\to X$
  a linear map with distinct eigenvalues
  $\lambda_1,\dots,\lambda_n$ and corresponding eigenvectors $v_1,\dots,v_n$.
  \begin{enumerate}
  \item Prove that $A$ has the same eigenvalues as the transpose map
    $A'\colon X'\to X'$.
  \item Let $\ell_1,\dots,\ell_n$ be the eigenvectors of $A'$. Prove
    that $(\ell_i,v_i)\neq 0$ for $i=1,\dots,n$.
  \item Explain why every $x\in X$ can be written as
    $x=a_1v_1+\cdots+a_nv_n$, and derive a formula for $a_i$.
  \item Find the dual basis of $v_1,\dots,v_n$. \bigskip
  \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
  \[
  p_A(s)=p_B(s)=p_C(s)=(s-2)^2(s-3)\,.
  \]
  \begin{enumerate}
  \item Find the eigenvectors and the minimal polynomials of each matrix
  \item Find a basis $\{v_1,v_2,v_3\}$ for $\R^3$ where $Bv_1=3v_1$,
    $Bv_2=2v_2$, and $(B-2I)v_3=v_2$. Write the matrix of this linear map
    with respect to this new basis. 
  \item Repeat the previous step for the matrix $C$.
  \end{enumerate}

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

\end{enumerate}

\end{document}







  
