# imaginary scator's involution

Submitted by mfg on Tue, 07/11/2017 - 11:42

## Conjugation

The main second order involution of a scator ${\overset {o}{ \varphi} } ={ f }_{ 0 }+\sum _{ j=1 }^{ n }{ { f }_{ j }{ \check { \mathbf{ e } }_{ j } }}$ is defined by the negative of its director components, while the scalar component remains unaltered ${ \overset{o}{\varphi} }^{ \ast }={ f }_{ 0 }-\sum _{ j=1 }^{ n }{ { f }_{ j }{ \check {\mathbf{ e } }_{ j } }}$.

## Magnitude

The square magnitude is defined by the product of a scator times its conjugate,  $\left\| \overset{o}{\varphi} \right\| ^{ 2 } =\overset{o}{\varphi} { \overset{o}{\varphi} }^{ \ast}$. For two or more director components, $$\left\| \overset{o}{\varphi} \right\| ^{ 2 }=\overset{o}{\varphi} { \overset{o}{\varphi} }^{ \ast }={ f }_{ 0 }^{ 2 }\prod _{ k=1 }^{ n }{ \left( 1+\frac { { f }_{ k }^{ 2 } }{ { f }_{ 0 }^{ 2 } } \right) },$$ whereas if there is only one possibly vanishing director component and a scalar component, say the ${ l }^{th}$ component, the square magnitude is$${ \left\| \overset{o}{\varphi} \right\| }^{ 2 }=\overset{o}{\varphi} { \overset{o}{\varphi} }^{ \ast }=\left( { f }_{ 0 }^{ 2 }+{ f }_{ l }^{ 2 } \right).$$

The magnitude is given by the positive square root  of the above expressions

$$\left\| \overset{o}{\varphi} \right\|=\bigl|f_{0}\bigr|\prod_{k=1}^{n}\sqrt{1+\frac{f_{k}^{2}}{f_{0}^{2}}}.\label{eq:a to m-vars scal}$$

## Multiplicative inverse

The multiplicative inverse can be obtained from the above results provided that the magnitude is not zero. Let $\overset{o}{\varphi}$ be an arbitrary scator, its inverse is ${ \overset{o}{\varphi} }^{ -1 }=\frac { { \overset{o}{\varphi} }^{ \ast } }{ \overset{o}{\varphi} { \overset{o}{\varphi} }^{\ast } }$. The denominator is the square magnitude $\overset{o}{\varphi}$, thus, if the square magnitude is written in terms of the scalar component and $n$ director components directoras $${ \overset{o}{\varphi} }^{ -1 }=\frac { { f }_{ 0 }-\sum _{ j=1 }^{ n }{ { f }_{ j }{ \check {\mathbf{ e } }_{ j } }} }{ { f }_{ 0 }^{ 2 }\prod _{ k=1 }^{ n }{ \left( 1+\frac { { f }_{ k }^{ 2 } }{ { f }_{ 0 }^{ 2 } } \right) } } .$$ If there is only one director component, say the ${ l }^{th}$  director, the multiplicative inverse is $${ \overset{o}{\varphi} }^{ -1 }=\frac { { f }_{ 0 }- {{ f }_{ l }{ \check {\mathbf{ e } }_{ l } }} }{ \left( { f }_{ 0 }^{ 2 }+{ f }_{ l }^{ 2 } \right) }.$$