Lecture13 - \documentclass[12pt,letterpaper]cfw_article

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
\documentclass[12pt,letterpaper]{article} \ \usepackage{amsmath} \usepackage{amssymb} \usepackage{latexsym} \usepackage{array} \usepackage{enumerate} \usepackage{amsthm} \usepackage{amscd} \ \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 13} \author{} \maketitle \ \textbf{\underline{Differentiability}} (of inversion)\\ \\ \ First suppose $F$ is differentiable at $ A\in G(X)$. From \[F(A)A = I\] we differentiate by the Leibniz rule and obtain \[F_A'(H)A + F(A)H = 0 \] \[F_A'(H) = - F(A)HA^{-1} = -A^{-1}HA^{-1}.\] This shows what the inverse is, assuming its existence. Now calculate, \[ F(A+H)- F(A)- \{-A^{-1}HA^{-1}\} = (A+H)^{-1} - A^{-1} + A^{-1}HA^{-1}\] \[ = ((I+HA^{-1})A)^{-1} - A^{-1}+ A^{-1}HA^{-1} \] \[ = A^{-1} \{ (I+HA^{-1})^{-1} - I + HA^{-1}. \} \] Now recall, N \[ (I+K)^{-1} = \sum_{n=0}^{\infty} (-K)^n \] \ \[ = I - K + \sum_{n=2}^{\infty}(-K)^n \] So \[\|(I+K)^{-1} -I + K \| \leq \|K\|^{2}\sum_{n=0}^{\infty} \|K\|^n\] \ \[= \frac{\|K\|^2}{1-\|K\|} \ (\|K\| < 1)\] \[= \frac{\|K\|^2}{1- \|K\|}\leq 2\|K\|^2 (\|K\| < \frac{1}{2}).\] \ Therefore, \[\|F(A+H)-F(A)+A^{-1}HA^{-1}\| \leq \|A^{-1}\| \|(I+HA^{-1})^{-1} - I + HA^{-1}\| \] \[ \leq \|A^{-1}\|2 \|HA^{-1}\|^2 \; \; ({\rm when} \|HA^{-1} \leq \frac{1}
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
{2};\; \; {\rm e.g. when} \|H\|\leq \frac{1}{2} \|A^{-1}\|^{-1}\|) \] Thus, if \[ \|H\| < \frac{1}{2} \|A^{-1}\|^{-1} \] then \[ F(A+H) - F(A) + A^{-1}HA^{-1}\| \leq 2\|A^{-1}\|^3\|H\|^2 \] and so \[ \frac{\|F(A+H)-F(A) + A^{-1}HA^{-1}\|}{\|H\|} \leq 2\|A^{-1}\|^3\|H\|\rightarrow 0 \; \; {\rm as} \|H\|\rightarrow 0. \] 2 \begin{theorem} If $X$ is a Banach space then \[F: G(X)\rightarrow
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

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.

Page1 / 6

Lecture13 - \documentclass[12pt,letterpaper]cfw_article

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online