From 4359ea153bdf6212f1ecd120306586c5e0411c97 Mon Sep 17 00:00:00 2001 From: hackbard Date: Thu, 9 Feb 2012 16:55:56 +0100 Subject: [PATCH] more math --- physics_compact/math_app.tex | 32 +++++++++++++++++++++++++++++++- physics_compact/phys_comp.tex | 1 + 2 files changed, 32 insertions(+), 1 deletion(-) diff --git a/physics_compact/math_app.tex b/physics_compact/math_app.tex index 001ed2b..9ff2b4b 100644 --- a/physics_compact/math_app.tex +++ b/physics_compact/math_app.tex @@ -40,9 +40,39 @@ The addition of two vectors is called vector addition. \label{math_app:product} \begin{definition} -The inner product ... +The inner product on a vector space $V$ over $K$ is a map $(\cdot,\cdot):V\times V \rightarrow K$, which satisfies +\begin{itemize} +\item $(\vec{u},\vec{v})=(\vec{v},\vec{u})^*$ + (conjugate symmetry, symmetric for $K=\mathbb{R}$) +\item $(\lambda\vec{u},\vec{v})=\lambda(\vec{u},\vec{v})$ and + $(\vec{u}'+\vec{u}'',\vec{v})=(\vec{u}',\vec{v})+(\vec{u}'',\vec{v})$ + (linearity in first argument) +\item $(\vec{u},\vec{u})\geq 0 \text{, } =" \Leftrightarrow \vec{u}=0$ + (positive definite) +\end{itemize} +for $\vec{u},\vec{v}\in V$ and $\lambda\in K$. \end{definition} +\begin{remark} +Due to conjugate symmetry, linearity in the first argument results in conjugate linearity (also termed antilinearity) in the second argument. +This is called a sesquilinear form. +$$+(\vec{u},\lambda(\vec{v}'+\vec{v}''))=(\lambda(\vec{v}'+\vec{v}''),\vec{u})^*= +\lambda^*(\vec{v}',\vec{u})^*+\lambda^*(\vec{v}'',\vec{u})^*= +\lambda^*(\vec{u},\vec{v}')+\lambda^*(\vec{u},\vec{v}'') +$$ +In physics and matrix algebra, the inner product is often defined with linearity in the second argument and conjugate linearity in the first argument. +This allows to express the inner product $(\vec{u},\vec{v})$ as a product of vector $\vec{v}$ with the dual vector or linear functional of dual space $V^{\dagger}$ +$$+(\vec{u},\vec{v}) \rightarrow \vec{u}^{\dagger}(\vec{u})\vec{v} +$$ +or the conjugate transpose in matrix formalism +$$+(\vec{u},\vec{v}) \rightarrow \vec{u}^{\dagger}\vec{v} \text{ .} +$$ +In doing so, conjugacy is associated with duality. +\end{remark} + \begin{definition} If $\vec{u}\in U$, $\vec{v}\in V$ and $\vec{v}^{\dagger}\in V^{\dagger}$ are vectors within the respective vector spaces and $V^{\dagger}$ is the dual space of $V$, the outer product $\vec{u}\otimes\vec{v}$ is defined as the tensor product of $\vec{v}^{\dagger}$ and $\vec{u}$, diff --git a/physics_compact/phys_comp.tex b/physics_compact/phys_comp.tex index eddfa8f..c82c627 100644 --- a/physics_compact/phys_comp.tex +++ b/physics_compact/phys_comp.tex @@ -8,6 +8,7 @@ \usepackage[latin1]{inputenc} \usepackage[T1]{fontenc} \usepackage{amsmath} +\usepackage{amssymb} \usepackage{ae} \usepackage{aecompl} \usepackage[dvips]{graphicx} -- 2.11.0