Lecture10

# Lecture10 -...

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Unformatted text preview: \documentclass[12pt,letterpaper]{article} \ \usepackage{amsmath} \usepackage{amssymb} \usepackage{latexsym} \usepackage{array} \usepackage{enumerate} \usepackage{amsthm} \usepackage{amscd} \ \newtheoremstyle{dotless}{}{}{}{}{\bfseries}{}{ }{} \newtheorem{lemma}{Lemma} \newtheorem{theorem}{Theorem} \newtheorem{definition}{Definition} \theoremstyle{dotless} \newtheorem*{remark}{Remark:} \newtheorem*{recall}{Recall:} \newtheorem*{fact}{Fact:} \ \newcommand{\bC}{{\mathbb C}} \newcommand{\bH}{{\mathbb H}} \newcommand{\bR}{{\mathbb R}} \newcommand{\ra}{\rightarrow} \newcommand{\Ra}{\Rightarrow} \DeclareMathOperator{\DD}{Det} \DeclareMathOperator{\TT}{Tr} \ \date{} \ \begin{document} \ \title{\bf Lecture 10} \author{} \maketitle \ \noindent \textbf{[Interlude}: \\ \\ $\DD_A ' (H) = (\DD A) \TT(A^{-1} H)$ if $A \in M_n (\bR)$ is invertible and $H \in M_n (\bR)$. M \begin{proof} Write $A = \left[ \begin{array}{ccccc} a_1 & \vline & \ldots & \vline & a_n \end{array} \right]$ with columns $a_1, \ldots , a_n \in \bR ^n$. In particular, $A = \left[ \begin{array} {ccccc} e_1 & \vline & \ldots & \vline & e_n \end{array} \right]$ for the standard basis $e_1, \ldots , e_n \mbox{ of } \bR ^n$. Since $\DD A$ is multilinear in $a_1, \ldots , a_n,$ $\DD$ is differentiable and $\DD_A ' (H) = \sum_{j=1}^n \DD\left[ \begin{array}{ccccccccc} a_1 & \vline & \ldots & \vline & h_j & \vline & \ldots & \vline & a_n \end{array} \right].$ \\ \emph{Special Case}: $A = I$. \begin{eqnarray*}...
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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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Lecture10 -...

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