Math235.tex

\documentclass{report}
\usepackage{xcolor,tikz}
\title{Math235 Notes}
\input{cp.tex}
\input{std.tex}
\input{defbox.tex}

\begin{document}

\maketitle
\tableofcontents
\chapter{Eigenvectors and Diagonalization}
\section{Similar Matrices}
\begin{quote}
Introducing Jennifer. She has her own language to represent any vector\\In her language, \{$\begin{bmatrix}1\\0\end{bmatrix}$,$\begin{bmatrix}0\\1\end{bmatrix}$\} equals \{$\begin{bmatrix}3\\2\end{bmatrix}$,$\begin{bmatrix}1\\-1\end{bmatrix}$\} in OUR Co-ordination System
\end{quote}

\begin{itemize}
\item Let $\mathbb B$ be the basis of \textbf{Jennifer's} Co-ordination System
\item Let L be the linear-transformation from \textbf{Jennifer's} Co-ordination System to \textbf{MY} Co-ordination System.
\item Let \textbf{Matrix A} be the standard Linear-Transformation of L $A=[L]$,in this case, $A=\begin{vmatrix}3&1\\2&-1\end{vmatrix}$
\item $[\vec x]_{\mathbb{B}}$ represent how \textbf{Jennifer} represent \ibx{$\vec x$ from \textbf{MY} Co-ordination System} using her language
\item Since we have P, \ibx{$\vec{x}$ in Jennifer's language} (i.e. $[\vec{x}]_{\mathbb{B}}$) multiple by P is $\vec{x}$ in Our System \ibx{$P[\vec{x}]_{\mathbb{B}}= \vec{x}$}
\item Inverse, $[\vec{x}]_\mathbb{B} = P^{-1}\vec{x}$,$(P^{-1})$ convert any vector in out language into Jennifer's language
\item
\end{itemize}
\chapter{Applications of Orthogonal Matrices}
\section{Orthogonal Similarity}
\subsection{Application}
\begin{itemize}
\item Find one of the real eigenvalues $C(\lambda )=\det(A-\lambda I)=0$
\item Find the corresponding eigenvector$(\vec v_1)$ $A-\lambda I \to{RREF}$
\item Extend $\vec v_1$ to orthonormal basis of $R^n$, Usually $I$, but not always
\item Calculate $P^T_1AP_1 = \begin{vmatrix}\lambda & \vec{b}^T\\\vec{0}&A_1\end{vmatrix}$
\item Inductively working on $A_1$,Notices that $\begin{vmatrix}\lambda & \vec{b}^T\\\vec{0}&A_1\end{vmatrix}$ is already upper triangular
\item We are looking for Orthogonal Matrix $Q$ such that $Q^TA_1Q = T_1$
\end{itemize}
\end{document}