lecture11 - \documentclass[12pt,letterpaper]cfw_article

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\documentclass[12pt,letterpaper]{article} \ \usepackage{amsmath} \usepackage{amssymb} \usepackage{latexsym} \usepackage{array} \usepackage{enumerate} \usepackage{amsthm} \usepackage{amscd} \usepackage{amsfonts} \ \newtheorem{lemma}{Lemma} \newtheorem{theorem}{Theorem} \newtheorem{definition}{Definition} \ \newcommand{\bC}{{\mathbb C}} \newcommand{\bH}{{\mathbb H}} \newcommand{\bR}{{\mathbb R}} \newcommand{\ra}{\rightarrow} \newcommand{\Ra}{\Rightarrow} \ \date{} \ \begin{document} \ \title{\bf Lecture 11} \author{} \maketitle \ \noindent \textbf{Interlude:} \\ \\ \ \begin{theorem} $ Det^{'}_{A}(H) = Tr(\widetilde{A}H) $ \end{theorem} \ \begin{proof} It is enough to see both sides agree when $H$ is a standard basis matrix $E_{ij} $. \\ Left Hand Side: \[ \begin{array}{rcl} &=& \frac{d}{dt}(DetA + (-1)^{i+j}tA_{ij})|_{t=0} \text { (where $A_{ij}$ is the minor of A)} \\ \\ m &=& (-1)^{i+j}A_{ij}. \end{array} \] Right Hand Side: \[ \begin{array}{rcl}
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&=& \sum^{n}_{r=1} B_{pr}\delta_{ir}\delta_{jq} \\ \\ \therefore Tr(BE_{ij})&=& \sum^{n}_{p=1} B_{pi}\delta_{jp} = B_{ji}. \\ \\ \text {In particular,}\ Tr(\widetilde{A}E_{ij}) &=& \widetilde{A}_{ji} = (- 1)^{i+j}A_{ij}. 1 \end{array} \] \end{proof} \ \medbreak \begin{lemma} Let $F:U\ra \bR$ where $U\subset X$ and $X$ is a normed space. Suppose $F$ is differentiable at $a\in U$ and has a local maximum value at $a$ then $F^{'}_{a} (h)=0$ for all $h$. (Exercise) \end{lemma} \begin{proof} Suppose BWOC there exists $x\in U$ such that $F^{'}_{a}(x)\neq 0$. \\ WLOG we can take $F^{'}_{a}(x)> 0$ (if $F^{'}_{a}(x)< 0$ look at $-x$). \\ Since $a$ is a local maximum, there exists $\delta_1$ such that
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This note was uploaded on 11/09/2011 for the course MAT 6932 taught by Professor Staff during the Spring '10 term at University of Florida.

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lecture11 - \documentclass[12pt,letterpaper]cfw_article

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