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\pfinal{}{Actual final time: 8-11am, Wednesday, March 19.}

\noindent This test is out of \textbf{20} points.  It is split into \textbf{two} main sections, with an additional bonus section at the end:
\begin{enumerate}
\item[$20$ points:]  A \textbf{calculational} section.  In this section, you are asked to calculate a series of objects related to what we've been studying in class. Do \textbf{all} of the problems in this section.
\item[$20$ points:] A \textbf{proof}.  In this section, you will choose \textbf{two} problems out of the three presented, and \textbf{prove} the claim it makes.  If you attempt more than one problem, the first problem encountered in the blue book will be graded.  Claims without proof will lose points when graded, so make sure to justify your answers.
\item[$+5$ points:] A \textbf{bonus} question!  This problem is fun/hard.
\end{enumerate}
\textbf{Resources allowed}: up to \textbf{four} hand-written sheets of paper, in your own handwriting.  Also, arbitrary amounts of blank paper, for scratch work.   No textbooks or other notes are allowed.  No calculators (they won't be helpful.)  When proving results, you are allowed to use anything that we've proven in class or in the HW without proof.  Other things must be proven if you want to use them.  It bears noting that every problem on this test can be solved using only things you have learned in this class and homework, and outside resources are borderline-useless.\\

\noindent\textbf{All work must be written in a blue book!}  As with the midterm, work not in a blue book will be used for decorative origami and in particular not graded.\\

\noindent \textbf{Good luck and have fun!}
\newpage

\section{Calculational Problems}
Do \textbf{all} of the problems here.  Justify your calculations: i.e.\ show the work that you used to derive your answers.  
\begin{enumerate}
\item Consider the matrix
\begin{align*}
A = \begin{bmatrix}
13 & -4 & 2\\
2 & 7 & 1\\
-4 & 4 & 7
\end{bmatrix}
\end{align*}
Find its Schur decomposition.

\item Consider the matrix  
\begin{align*}
B = \begin{bmatrix}
11 & 1 & 1 & 1\\
1 & 11 & 1 & 1\\
1 & 1 & 11 & 1\\
1 & 1 & 1 & 11\\
\end{bmatrix}
\end{align*}
Find a diagonal matrix $D$ that $B$ is similar to.

\item Consider the matrix  
\begin{align*}
C = \begin{bmatrix}
2 & 2 & 0 & 0\\
2 & 2 & 0 & 0\\
0 & 0 & 2 & 2\\
0 & 0 & 2 & 2\\
\end{bmatrix}
\end{align*}
Find a block-diagonal matrix $F$ whose blocks are Jordan blocks that $C$ is similar to.

\item Take the matrix 
\begin{align*}
G = \begin{bmatrix}
1 & 4 & 2\\
0 & 3 & 5\\
0 & 0 & 3\\
\end{bmatrix}
\end{align*}
Find elementary matrices $E_1, E_2, E_3$ such that $E_1( E_2( E_3 G E_3^{-1}) E_2^{-1}) E_1^{-1}$ is a matrix in Jordan normal form.
\end{enumerate}
\newpage

\section{Proof-Based Problems}

Choose \textbf{two} of the three problems below.  If you answer three problems, we will grade the first two that appear in your solutions.

\begin{enumerate}
\item Suppose that $A$ is a $n \times n$ matrix with the following two properties:
\begin{itemize}
\item $A^n$ is the all-zeroes matrix.
\item There is exactly one nonzero vector $\vec{v}$, up to scalar multiples, that is an eigenvector of $A$.  (In other words, the only eigenvectors for $A$ are vectors of the form $c\cdot \vec{v}$.)
\end{itemize}
Find the Jordan normal form of $A$.

\item Suppose that $A$ is a real-valued symmetric $n \times n$ matrix with the following two properties:
\begin{itemize}
\item All of $A$'s entries are either 0 or 1.
\item The all-1's vector is an eigenvector of $A$, with eigenvalue $10$.
\end{itemize}
\begin{enumerate}
\item How many 1's are there in each row of $A$?  
\item Suppose that $\lambda \neq 10$ is another eigenvalue of $A$.  Prove that $\lambda \leq 10$. 
\end{enumerate}

\item Suppose that $P$ is a $n \times n$ permutation matrix.  Show that the determinant of $P^2$ is 1.
\end{enumerate}
\newpage
\section{Bonus Problem}
\begin{enumerate}
\item A disgruntled electrical engineer, on his last day of work, decided to rewire all of the connections between the light switches at his job.  Now, each switch is connected to several different lights, and flicking that switch changes the states of all of those lights; if they are on, they become off, and vice-versa.  

After some experimentation, you have discovered the following fact: given any set of lights in the building, there is a switch that is connected to an odd number of lights in that set (as well as possibly other lights not in that set.)  

Prove that there is some way to turn off all of the lights.
\end{enumerate}


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