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\begin{center}
  {\large Mathematics 111}\\Spring 2009\\Homework 6
\end{center}

\begin{enumerate}

\item Let $f$ be a monic polynomial of degree $n \ge 1$, and let
  $C(f)$ be the companion matrix.  
  \begin{enumerate}
  \item Although we have shown in class that
  the minimal polynomial of $C(f)$ is $f$, give a proof (independent
  of this fact) that the characteristic polynomial of $C(f)$ is $f$.

\item Let $T$ be a linear operator on a finite dimensional vector
  space $V$ over a field $k$, and let $q_1 \mid q_2 \mid \dots \mid
  q_s$ be the invariant factors associated to $V$ as a torsion
  $k[x]$-module. Show that $q_1q_2\cdots q_s = c_T$, where $c_T$ is
  the characteristic polynomial of $T$.  Show that the minimal
  polynomial of $T$ divides the characteristic polynomial.  Note that
  this proves the Cayley-Hamilton theorem.  \textit{Hint:}  You may
  use without proof (though you should think about it) that the
  determinant of a matrix of the form 
  $\begin{pmatrix}
    A&B\\0&C
  \end{pmatrix}$ with $A$ and $C$ square matrices is the product
  $\det(A)\det(C)$.

  \end{enumerate}

\item Find all rational and Jordan canonical forms of a matrix in
  $M_5(\C)$ having minimal polynomial $x^2(x-1)$.  Be sure to give the
  corresponding invariants and the characteristic polynomials.


\item Show that any linear operator $T$ on a finite dimensional vector
  space (over a field of characteristic not equal to 2) which
  satisfies $T^2 = I$ is diagonalizable.  Give all possible Jordan
  forms for $4\times 4$ matrices $A$ with $A^2 = I$.

\item Consider a matrix of the form
  $A = \begin{pmatrix}
    \lambda&\mu&0&\dots&0\\
    0&\lambda&\mu&\dots&0\\
    0&0&\ddots&\ddots&0\\
    0&0&\dots&\lambda&\mu\\
    0&0&\dots&0&\lambda\\
  \end{pmatrix}$, i.e., with diagonal $\lambda$ and $\mu$ on the
  superdiagonal. Find the Jordan canonical form(s) of $A$.  The answer
  should depend slightly on $\mu$.  What does your answer say about
  the form of Jordan blocks as introduced in the text in comparison
  with the way we defined them?


  
\item Let $T$ be a linear operator on a finite dimensional vector
  space $V$.  Let $\lambda_1,\dots, \lambda_k$ be distinct eigenvalues of
  $T$ and let $v_1, \dots, v_k$ be corresponding (non-zero)
  eigenvectors.  Show that $\{v_1, \dots, v_k\}$ is linearly
  independent.  \textit{Hint:}  Apply $T-\lambda_k I$ to a dependence
  relation and use induction.
\end{enumerate}






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