\documentclass[11pt]{article}
\usepackage{amsfonts}
\usepackage{amsthm}
\usepackage{amsmath}
\usepackage{latexsym}
\usepackage{multicol}
\usepackage{verbatim}
\usepackage[pdftex]{graphicx}
\usepackage{url}
\usepackage{cancel}
\usepackage[table]{xcolor}
\usepackage[all,arc]{xy}
\usepackage{color, colortbl}
\usepackage{mathrsfs}
\usepackage{hyperref}
\hypersetup{colorlinks=true}
\setlength{\oddsidemargin}{.25in}
\setlength{\evensidemargin}{.25in}
\setlength{\textwidth}{6in}
\setlength{\topmargin}{-0.4in}
\setlength{\textheight}{8.5in}

\input{preamble.tex}

\begin{document}

\homework{6: Similar/Elementary Matrices; the Spectral Theorem}{Due Wednesday, February 26, by 1:30pm}


Homework problems need to show work and contain proofs in order to receive full credit.  Simply stating an answer is only half of the problem in mathematics; you also need to include an argument that persuades your audience that your answer is correct!  If you have any questions, feel free to contact either Yihan or I via email or office hours.  Have fun!

\begin{enumerate}
\item Suppose that $A$ and $B$ are a pair of $2 \times 2$ matrices, such that $\lambda$ is an eigenvalue of $A$ if and only if it is an eigenvalue of $B$.  Is $A$ similar to $B$?  Either create a counterexample to this claim, or prove that it is true.
\item \begin{enumerate}
\item Prove that if $A$ and $B$ are similar, then the trace of $A$ is the trace of $B$.
\item Prove that if $A$ and $B$ are similar, then the determinant of $A$ is the determinant of $B$.
\end{enumerate}
\item \begin{enumerate}
\item Prove the following claim: the only matrix that is similar to the identity matrix is the identity matrix.
\item Is there any other matrix $A$ such that the only matrix similar to $A$ is $A$ itself?  Come up with a very simple property characterizing all such matrices.  Prove that your property indeed characterizes these kinds of matrices: i.e. prove that any matrix that satisfies your property is only similar to itself, and that if a matrix is only similar to itself then it satisfies your property.
\end{enumerate}
\item Consider the following elementary matrix:
\begin{align*}
E = \begin{bmatrix}
1 & 0 & 0\\
0 & 1 & 0\\
\lambda & 0 & 1\\
\end{bmatrix}
\end{align*}
Find an upper-triangular matrix that is similar to $E$. 
\item[(Bonus!)]   If you solve this problem, email your answers to me directly; answers sent to the TA only will not be graded.\\  Take the field $\mathbb{Z} / 2 \mathbb{Z}  = \langle \{0, 1\}, +, \cdot \rangle$, given by the usual concept of modular arithmetic mod 2.  With this field, we can define vectors in $(\mathbb{Z}/2\mathbb{Z})^n$, like $(0,1) \in (\mathbb{Z}/2\mathbb{Z})^2$, and matrices like $\begin{bmatrix} 1 & 1 \\0 & 0 \end{bmatrix}$ that act on these vectors; as well, we can define the concepts of matrix multiplication mod 2 (i.e.\ $\begin{bmatrix} 1 & 1 \\ 0 & 1\end{bmatrix} \cdot \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix} \equiv \begin{bmatrix}0 & 1\\ 1 & 0 \end{bmatrix} \mod 2$.) For any $n$, call the collection of $n \times n$ matrices with entries from this field $M_n(\mathbb{Z}/2\mathbb{Z})$.

For any $n$, what is the largest set of $n \times n$ matrices in $M_n(\mathbb{Z}/2\mathbb{Z})$ that you can find, such that no two of these matrices are similar via some third element in $M_n(\mathbb{Z}/2\mathbb{Z})$?
\end{enumerate}





\end{document}