From: hackbard <hackbard@hackdaworld.org>
Date: Thu, 9 Feb 2012 15:55:56 +0000 (+0100)
Subject: more math
X-Git-Url: https://hackdaworld.org/gitweb/?a=commitdiff_plain;h=4359ea153bdf6212f1ecd120306586c5e0411c97;p=lectures%2Flatex.git

more math
---

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.
+\begin{equation}
+(\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}'')
+\end{equation}
+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 $V^{\dagger}$
+\begin{equation}
+(\vec{u},\vec{v}) \rightarrow \vec{u}^{\dagger}(\vec{u})\vec{v}
+\end{equation}
+or the conjugate transpose in matrix formalism
+\begin{equation}
+(\vec{u},\vec{v}) \rightarrow \vec{u}^{\dagger}\vec{v} \text{ .}
+\end{equation}
+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}