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\begin{document}
\begin{center}
{\bf Homework 11}\\
(due Wednesday, May 2)
\end{center}

\vspace*{.3in}
{\noindent\bf I've fixed some typos and added a problem since first posting this homework.  The total number of problems remains the same, because I also removed the old problem 2.}
\vspace*{.3in}

\begin{prob}
Let $V$ be a vector space over $\F$, $T:V\to V$ be a linear operator and $p\in\F[x]$ an irreducible polynomial.  Let 
$$
H_p := \{\vv\in V: p(T)^k\vv = \vzero \text{ for some } k\in\N\}.
$$
\begin{enumerate}
 \item Show directly from the definition of $H_p$ that $H_p$ is a $T$-invariant subspace of $V$.
 \item Show that in fact $H_p = \ker p(T)^r$ where $r\in\N$ is the largest integer such that $p^r$ divides the minimal polynomial $p_{min}$ of $T$.
 \item Infer that $H_p$ is non-trivial if and only if $p|p_{min}$. 
 \item Show that if $P\in\F[x]$ is any polynomial, irreducible or not, then $P(T)$ is invertible if and only if $\gcd(P,p_{min}) = 1$, and in this case there is another polynomial $Q\in\F[x]$ such that $P(T)^{-1} = Q(T)$.
\end{enumerate}
Avoid using the primary decomposition theorem to do any of this.
\end{prob}

\begin{prob}
Since all entries of the matrix
$$
A = \left[\begin{matrix} 0 & 1 & -2 & 9 \\ -1 & 2 & -1 & 4 \\ 2 & -1 & 4 & -11 \\ 1 & 0 & 1 & -2\end{matrix}\right],
$$
are integers, $A$ defines a linear operator $T:\F^4\to\F^4$, where one can take $\F$ to be \emph{any} field--e.g. $\C$, $\R$, or $\Q$.
Feel free to use your calculator or computer to help with the following.
\begin{enumerate}
 \item The minimal and characteristic polynomials for $T$ are equal and do not depend on the field $\F$.  Find them by computing $p_{T,\vv}$ for a ``suitably random'' vector $\vv$.
 \item Let $T:\Q^4 \to \Q^4$ denote the linear transformation $T(\vx) = A\vx$ given by $A$.  Describe (i.e. give bases for each of the subspaces in) the primary decomposition of $\Q^4$ relative to $T$.
 \item Let $\cB$ be the basis for $\Q^4$ obtained by taking the union of the bases found in the previous part of this problem.  Give the matrix for $T$ relative to $\cB$.
 \item How does your answer to the previous two parts of this problem change we take $\F=\R$; i.e. if $T$ is regarded as a linear operator on $\R^4$?
 \item And what if we take $\F=\C$?
 \item Describe the $SN$ decomposition of $A$---i.e. write $A = S+N$ where $S$ and $N$ are commuting matrices such that $S$ is diagonalizable and $N$ is nilpotent.
 \item Find $e^A$
\end{enumerate}
\end{prob}

\begin{prob} (The bonus problem from last semester's final)
Suppose that $V$ is a finite dimensional real vector space and $T:V\to V$ is a linear transformation such that $T^2 = \id$.  Prove that $T$ is diagonalizable and that every eigenvalue of $T$ is either $1$ or $-1$.  In fact, do it twice: once using the primary decomposition theorem and once ``directly'' without recourse to any recent theorems from class.
\end{prob}

\begin{prob}
Let $C^\infty$ be the vector space of infinitely differentiable complex-valued functions $y:\R\to\C$.  Let $D:C^{\infty}\to C^{\infty}$ denote the differentiation operator $Dy = y'$.  Recall that a homogeneous linear differential equation is one that can be written
$$
P(D)y = 0,
$$
for some polynomial $P(x) = (x-\lambda_1)^{m_1}\dots (x-\lambda_k)^{m_k} \in\C[x]$.  We also learned in class that the set $V := \ker P(D)$ of solutions is a subspace of $C^\infty$ with dimension equal to $\deg P$.  In particular, the restriction $D:V\to V$ is a linear operator on a finite dimensional complex vector space.  
\begin{itemize}
 \item What is the primary decomposition of $D|_V$?  Use what you know about solutions of linear DEs to give (explicit) bases for each primary subspace.
 \item What are the characteristic and minimal polynomials of $D|_V$?  Why?
 \item What is the Jordan form for $D|_V$?  That is, give a basis $\cB$ for $V$ such that $[D]_{\cB\cB}$ is a matrix in Jordan form and then describe the matrix. 
\end{itemize}
\end{prob}






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