From dcfe9181a03b1eff2ec41d207241647e7392f2fd Mon Sep 17 00:00:00 2001 From: hackbard Date: Thu, 9 Feb 2012 21:49:56 +0100 Subject: [PATCH] fixed outer product, redo inner product! --- physics_compact/math_app.tex | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/physics_compact/math_app.tex b/physics_compact/math_app.tex index 9ff2b4b..79e4ec9 100644 --- a/physics_compact/math_app.tex +++ b/physics_compact/math_app.tex @@ -74,14 +74,14 @@ 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 $\vec{u}$, +If $\vec{u}\in U$, $\vec{v}\in V$ are vectors within the respective vector spaces and $\vec{y}^{\dagger}\in V^{\dagger}$ is a linear functional of the dual space $V^{\dagger}$ of $V$, +the outer product $\vec{u}\otimes\vec{v}$ is defined as the tensor product of $\vec{y}^{\dagger}$ and $\vec{u}$, which constitutes a map $A:V\rightarrow U$ by \begin{equation} -\vec{v}\mapsto\vec{v}^{\dagger}(\vec{v})\vec{u} +\vec{v}\mapsto\vec{y}^{\dagger}(\vec{v})\vec{u} \text{ ,} \end{equation} -where $\vec{v}^{\dagger}(\vec{v})$ denotes the linear functional $\vec{v}^{\dagger}\in V^{\dagger}$ on $V$ when evaluated at $\vec{v}\in V$, a scalar that in turn is multiplied with $\vec{u}\in U$. +where $\vec{y}^{\dagger}(\vec{v})$ denotes the linear functional $\vec{y}^{\dagger}\in V^{\dagger}$ on $V$ when evaluated at $\vec{v}\in V$, a scalar that in turn is multiplied with $\vec{u}\in U$. In matrix formalism, with respect to a given basis ${\vec{e}_i}$ of $\vec{u}$ and ${\vec{e}'_i}$ of $\vec{v}$, if $\vec{u}=\sum_i^m \vec{e}_iu_i$ and $\vec{v}=\sum_i^n\vec{e}'_iv_i$, -- 2.20.1