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\pagestyle{fancy} \lhead{{\sf Homework 9 $|$ Due March 27
    (Wednesday) }} \rhead{\thepage} \cfoot{{\sf MthSc 853 $|$ Linear
    Algebra $|$ Spring 2013 $|$ M.~Macauley}}


\begin{document}

$\;$

\noindent\emph{Read}: Lax, Appendix 15, pages 363--366. \\

\begin{enumerate}


% 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?
  \end{enumerate}



\item Compute the Jordan canonical form of the following matrices, and
  give a basis of genuine and generalized eigenvectors for each.
  \[
  A=\begin{bmatrix}-1&0&1&0\\2&1&2&1\\0&0&-1&0\\4&0&-6&1\end{bmatrix}
  \,,\qquad\qquad
  B=\begin{bmatrix}1&0&0&1\\2&1&0&-4\\1&0&1&-2\\0&0&0&1\end{bmatrix}\,.
  \]



% Lax Exercise 6.8, modified
\item Let $A$ be a matrix with distinct eigenvalues
  $\lambda_1,\dots,\lambda_k$, and denote the index of $\lambda_i$ by
  $d_i$.
  \begin{enumerate}
  \item Prove, without using Jordan canonical form, that the minimal
    polynomial of $A$ is
    \[
    m_A(t)=\prod_{i=1}^k(t-\lambda_i)^{d_i}\,.
    \]
  \item Give a simple proof using the Jordan canoncial form.
  \end{enumerate}



\item Find a list of real matrices, as long as possible, such that
  \begin{enumerate}[(i)]
  \item The characteristic polynomial of each matrix is $(t-1)^5(t+1)$
  \item The minimal polynomial of each matrix is $(t-1)^2(t+1)$
  \item No two matrices in the list are similar to each other.
  \end{enumerate}



\item Let $X\subset\R[x,y]$ be the space of polynomials in $x$, $y$ of
  total degree $\leq n$. Find the dimension of $X$, show that the map
  \[
  A:X\longrightarrow X,\qquad f\longmapsto f+\frac{\partial f}{\partial
    x}+\frac{\partial f}{\partial y}
  \]
  is linear, and find its Jordan canonical form. 



\item Consider the following matrix:
  \[
  M_n=\begin{bmatrix}0&-a_0\\I_{n-1}&-\textbf{a}_{n-1}\end{bmatrix}
  \quad\text{where}\;\;
  \textbf{a}_n=\begin{bmatrix}a_1\\a_2\\\vdots\\a_n\end{bmatrix}.
  \]
  \begin{enumerate}
  \item Show that the characteristic polynomial of $M_n$ is
    \[
    P_{M_n}(t)=t^n+a_{n-1}t^{n-1}+\dots+a_1t+a_0\,. 
    \]
    Here, $I_{n-1}$ denotes the $(n-1)\times(n-1)$ identity matrix.
  \item Is $P_{M_n}(t)$ also the minimal polynomial? Prove or
    disprove.
  \item Now, let $X$ be a 4 dimensional vector space over $\R$ with
    basis $\{x_1,x_2,x_3,x_4\}$ and let $T:X\rightarrow X$ be a linear
    map such that
    \[
    T(x_1)=x_2,\quad T(x_2)=x_3,\quad T(x_3)=x_4,\quad
    T(x_4)=-x_1-4x_2-6x_3-4x_4. 
    \]
  \item Is $T$ diagonalizable over $\C$? Why or why not?
  \end{enumerate}


\end{enumerate}

\end{document}







  
