Entries for July 22, 2017

Equipping a vector space with an inner product results in a natural isomorphism $\CV\to\CV^\ast$ , where the metric tensor can be interpreted as the linear mapping $\Bg:\CV\to\CV^\ast$ and its inverse $\Bg\inv:\CV^\ast\to\CV$ .

Notation: Given two real vector spaces $\CV$ and $\CW$ , we denote their inner products as $\dabrn{\cdot,\cdot}_{\CV}$ and $\dabrn{\cdot,\cdot}_{\CW}$ respectively. Given vectors $\Bv\in\CV$ and $\Bw\in\CW$ , we define their lengths as

\begin{equation} \Norm{\Bv}_{\CV} = \sqrt{\dabrn{\Bv,\Bv}_{\CV}} \eqand \Norm{\Bw}_{\CW} = \sqrt{\dabrn{\Bw,\Bw}_{\CW}}. \tag{1}\end{equation}

Regarding $\CV$ and $\CW$ ,

their bases are denoted $\cbrn{\BE_A}$ and $\cbrn{\Be_a}$ ,
their dual bases are denoted $\cbrn{\BE^A}$ and $\cbrn{\Be^a}$ ,
their metrics are denoted $\BG$ and $\Bg$ with the components $G_{AB}=\dabrn{\BE_A,\BE_B}_{\CV}$ and $g_{ab}=\dabrn{\Be_a,\Be_b}_{\CW}$ ,

respectively. Here, the indices pertaining to $\CV$ are uppercase $(ABC\dots)$ and the indices pertaining to $\CW$ are lowercase $(abc\dots)$ .

Definition: Let $\BP:\CV\to\CW$ be a linear mapping. Then the transpose, or adjoint of $\BP$ , written $\BP\tra$ , is the linear mapping

\begin{equation} \boxed{ \BP\tra: \CW\to\CV \quad\text{such that}\quad \dabrn{\Bv,\BP\tra\Bw}_{\CV} = \dabrn{\BP\Bv,\Bw}_{\CW} } \tag{2}\end{equation}

for all $\Bv\in\CV$ and $\Bw\in\CW$ . Carrying out the products,

\begin{equation} G_{BA} v^B (P\tra){}^{A}{}_{d} w^d = g_{ab} P{}^{b}{}_{C}v^Cw^a. \tag{3}\end{equation}

For arbitrary $\Bv$ and $\Bw$ ,

\begin{equation} G_{BA} (P\tra){}^{A}{}_{a} = g_{ab} P{}^{b}{}_{A} \tag{4}\end{equation}

from which we can obtain the components of the transpose as

\begin{equation} \boxed{ (P\tra){}^{A}{}_{a} = g_{ab} P{}^{b}{}_{B} G^{AB} \eqwith \BP\tra = (P\tra){}^{A}{}_{a} \BE_A\dyd\Be^a . } \tag{5}\end{equation}

If $\BB:\CV\to\CV$ is a linear mapping, it is called symmetric if $\BB=\BB\tra$ .

Definition: Let $\BP:\CV\to\CW$ be a linear mapping. Then the dual of $\BP$ is a metric independent mapping

\begin{equation} \boxed{ \BP^\ast: \CW^\ast\to\CV^\ast \quad\text{such that}\quad \abrn{\Bv,\BP^\ast\Bbeta}_{\CV} = \abrn{\BP\Bv,\Bbeta}_{\CW} } \tag{6}\end{equation}

defined through natural pairings for all $\Bv\in\CV$ and $\Bbeta\in\CW^\ast$ . Carrying out the products,

\begin{equation} v^A (P^\ast){}_{A}{}^{a} \beta_a = P{}^{b}{}_{B} v^B \beta_b. \tag{7}\end{equation}

For arbitrary $\Bv$ and $\Bbeta$ , we obtain the components of the dual mapping as

\begin{equation} \boxed{ (P^\ast){}_{A}{}^{a} = P{}^{a}{}_{A} \eqwith \BP^\ast = (P^\ast){}_{A}{}^{a} \BE^A\dyd\Be_a = P{}^{a}{}_{A} \BE^A\dyd\Be_a . } \tag{8}\end{equation}

To fully appreciate the symmetry that originates from the duality, we can think of not just the mappings between $\CV$ and $\CW$ , but also between their dual spaces. To this end we can enumerate four mappings corresponding to $\cbr{\CV,\CV^\ast}\to\cbr{\CW,\CW^\ast}$ and their duals, corresponding to $\cbr{\CW,\CW^\ast}\to\cbr{\CV,\CV^\ast}$ . Their definitions can be found in the table below.

Tensors $\BP$ , $\BQ$ , $\BR$ and $\BS$ as linear mappings (top), and their duals $\BP^\ast$ , $\BQ^\ast$ , $\BR^\ast$ and $\BS^\ast$ (bottom). In the respective tables, the first row displays the tensor spaces, basis vectors and components of the subsequent mappings, and the second and third row display the representations of the tensor as linear and bilinear mappings respectively. The results of the mappings are given in the mapping, matrix and index representations respectively. The mappings are over vectors $\Bv\in\CV$ , $\Bw\in\CW$ and one-forms $\Balpha\in\CV^\ast$ , $\Bbeta\in\CW^\ast$ .
$\BP\in\CW \dyd\CV^\ast$ $P^a_{\idxsep A}=\BP(\Be^a,\BE_A)$ $\BP = P^{a}_{\idxsep A}\, \Be_a \dyd \BE^A$	$\BQ\in\CW^\ast \dyd\CV^\ast$ $Q_{aA}=\BQ(\Be_a,\BE_A)$ $\BQ = Q_{aA}\, \Be^a \dyd \BE^A$	$\BR\in\CW \dyd\CV$ $R^{aA}=\BR(\Be^a,\BE^A)$ $\BR = R^{aA}\, \Be_a \dyd \BE_A$	$\BS\in\CW^\ast \dyd\CV$ $S_a^{\idxsep A}=\BS(\Be_a,\BE^A)$ $\BS = S_{a}^{\idxsep A}\, \Be^a \dyd \BE_A$
Mappings
$\BP: \CV \to \CW$ $\begin{aligned} \Bv &\mapsto \BP(\Be^a,\Bv) \Be_a \\ &= \BP\Bv \\ v^A\BE_A &\mapsto P^a_{\idxsep A} v^A \Be_a \end{aligned}$	$\BQ: \CV \to \CW^\ast$ $\begin{aligned} \Bv &\mapsto \BQ(\Be_a,\Bv) \Be^a \\ &= \BQ\Bv \\ v^A\BE_A &\mapsto Q_{aA} v^A \Be^a \end{aligned}$	$\BR: \CV^\ast \to \CW$ $\begin{aligned} \Balpha &\mapsto \BR(\Be^a,\Balpha) \Be_a \\ &= \BR\Balpha\tra \\ \alpha_A \BE^A &\mapsto R^{aA} \alpha_A \Be_a \end{aligned}$	$\BS: \CV^\ast \to \CW^\ast$ $\begin{aligned} \Balpha &\mapsto \BS(\Be_a,\Balpha) \Be^a \\ &= \BS\Balpha\tra \\ \alpha_A \BE^A &\mapsto S_a^{\idxsep A} \alpha_A \Be^a \end{aligned}$
$\BP: \CW^\ast \times \CV \to \IR$ $\begin{aligned} (\Bbeta,\Bv) &\mapsto \BP(\Bbeta,\Bv) \\ &=\Bbeta\BP\Bv \\ &= \beta_a P^{a}_{\idxsep A} v^A \end{aligned}$	$\BQ: \CW \times \CV \to \IR$ $\begin{aligned} (\Bw,\Bv) &\mapsto \BQ(\Bw,\Bv) \\ &= \Bw\tra\BQ\Bv \\ &= w^a Q_{aA} v^A \end{aligned}$	$\BR: \CW^\ast \times \CV^\ast \to \IR$ $\begin{aligned} (\Bbeta,\Balpha) &\mapsto \BR(\Bbeta,\Balpha) \\ &= \Bbeta\BR\Balpha\tra \\ &= \beta_a R^{aA} \alpha_A \end{aligned}$	$\BS: \CW \times \CV^\ast \to \IR$ $\begin{aligned} (\Bw,\Balpha) &\mapsto \BS(\Bw,\Balpha) \\ &= \Bw\tra\BS\Balpha\tra \\ &= w^a S_a^{\idxsep A} \alpha_A \end{aligned}$
Duals
$\BP^\ast\in \CV^\ast \dyd\CW$ $P^{\ast \, a}_A=\BP^\ast(\BE_A,\Be^a)$ $\BP^\ast = P^{\ast \, a}_A \, \BE^A \dyd \Be_a$	$\BQ^\ast\in \CV^\ast \dyd\CW^\ast$ $Q^\ast_{Aa}=\BQ^\ast(\BE_A,\Be_a)$ $\BQ^\ast = Q^\ast_{Aa}\, \BE^A \dyd \Be^a$	$\BR^\ast\in\CV \dyd \CW$ $R^{\ast Aa}=\BR^\ast(\BE^A,\Be^a)$ $\BR^\ast = R^{\ast Aa}\, \BE_A \dyd \Be_a$	$\BS^\ast\in\CV \dyd\CW^\ast$ $S^{\ast A}{}_{a}=\BS^\ast(\BE^A,\Be_a)$ $\BS^\ast = S^{\ast A}{}_{a}\, \BE_A \dyd \Be^a$
$\BP^\ast: \CW^\ast \to \CV^\ast$ $\begin{aligned} \Bbeta &\mapsto \BP^\ast(\BE_A,\Bbeta) \BE^A \\ &= \BP^\ast\Bbeta\tra \\ \beta_a\Be^a &\mapsto P^{\ast \, a}_A \beta_a \BE^A \end{aligned}$	$\BQ^\ast: \CW \to\CV^\ast$ $\begin{aligned} \Bw &\mapsto \BQ^\ast(\BE_A,\Bw) \BE^A \\ &= \BQ^\ast\Bw \\ w^a\Be_a &\mapsto Q^\ast_{Aa} w^a \BE^A \end{aligned}$	$\BR^\ast: \CW^\ast \to \CV$ $\begin{aligned} \Bbeta &\mapsto \BR^\ast(\BE^A,\Bbeta) \BE_A \\ &= \BR^\ast\Bbeta\tra \\ \beta_a\Be^a &\mapsto R^{\ast Aa} w^a \BE_A \end{aligned}$	$\BS^\ast: \CW \to\CV$ $\begin{aligned} \Bw &\mapsto \BS^\ast(\BE^A,\Bw) \BE_A \\ &= \BS^\ast\Bw \\ w^a\Be_a &\mapsto S^{\ast A}{}_{a} w^a \BE_A \end{aligned}$
$\BP^\ast: \CV \times \CW^\ast \to \IR$ $\begin{aligned} (\Bv,\Bbeta) &\mapsto \BP^\ast(\Bv,\Bbeta) \\ &= \Bv\tra\BP^\ast\Bbeta\tra \\ &= v^A P^{\ast \, a}_A \beta_a \end{aligned}$	$\BQ^\ast: \CV \times \CW \to \IR$ $\begin{aligned} (\Bv,\Bw) &\mapsto \BR^\ast(\Bv,\Bw) \\ &= \Bv\tra\BQ^\ast\Bw \\ &= v^A Q^\ast_{Aa} w^a \end{aligned}$	$\BR^\ast: \CV^\ast \times \CW^\ast \to \IR$ $\begin{aligned} (\Balpha,\Bbeta) &\mapsto \BS^\ast(\Balpha,\Bbeta) \\ &= \Balpha\BR^\ast\Bbeta\tra \\ &= \alpha_A R^{\ast Aa} \beta_a \end{aligned}$	$\BS^\ast: \CV^\ast \times \CW \to \IR$ $\begin{aligned} (\Balpha,\Bw) &\mapsto \BS^\ast(\Balpha,\Bw) \\ &= \Balpha\BS^\ast\Bw \\ &= \alpha_A S^{\ast A}{}_{a} w^a \end{aligned}$

The commutative diagrams pertaining to these mappings can be found in the figure below

Commutative diagrams involving the linear mappings

\BP,\BQ,\BR,\BS

and their dual

\BP^\ast,\BQ^\ast,\BR^\ast,\BS^\ast

based on the metrics

\BG

and

\Bg

\CV

and

\CW

Isomorphisms in Linear Mappings between Vector Spaces