# real scators' involution

Submitted by mfg on Tue, 07/11/2017 - 18:29

## Conjugation

The main second order involution of the real scator ${ \varphi } ={ f }_{ 0 }+\sum _{ j=1 }^{ n }{ { f }_{ j }{ \hat { \mathbf{ e } }_{ j } }}$ is defined by the negative of his direction components, while the scalar component stands without changes ${ \varphi }^{ \ast }={ f }_{ 0 }-\sum _{ j=1 }^{ n }{ { f }_{ j }{ \hat {\mathbf{ e } }_{ j } }}$. The  ${ j }^{ ésimo }$ director conjugated of a scator $\varphi =\left( { f }_{ 0 };{ f }_{ 1 },...,{ f }_{ j },...,{ f }_{ n } \right)$, is labeled with an asterisk and is defined as the negative of the   ${ j }^{ ésima}$  direction component , while all other stands without changes ${ \varphi }^{ \ast j }\equiv \left( { f }_{ 0 };{ f }_{ 1 },...,{ -f }_{ j },...,{ f }_{ n } \right)$.

## Magnitude

the square of the magnitude of a writer is defined as the product  of the same with his conjugate, namely, $\left\| \varphi \right\| ^{ 2 } =\varphi { \varphi }^{ \ast}$ , then, for two or more direction components of the scator in question, the operation is defined as follows:\begin{equation}\left\| \varphi  \right\| ^{ 2 }=\varphi { \varphi  }^{ \ast  }={ f }_{ 0 }^{ 2 }\prod _{ k=1 }^{ n }{ \left( 1-\frac { { f }_{ k }^{ 2 } }{ { f }_{ 0 }^{ 2 } }  \right)  }, \end{equation} if only one has the scalar component and a single direction, namely, the ${ l }^{ ésima }$ component, so, the square of the magitude is:\begin{equation}{ \left\| \varphi  \right\|  }^{ 2 }=\varphi { \varphi  }^{ \ast }=\left( { f }_{ 0 }^{ 2 }-{ f }_{ l }^{ 2 } \right).\end{equation}

## Inverse multiplicative

Through the above-mentioned definitions, namely, the conjugate and the square magnitude of the scator, we can defined his multiplicative inverse. If  $\varphi$ is a scator, then, his inverse is ${ \varphi }^{ -1 }=\frac { { \varphi }^{ \ast } }{ \varphi { \varphi }^{\ast } }$, we can see that the term in the denominator  is the square of the magnitude of  $\varphi$, thus, if we replace the denominator the multiplicative inverse is as follows: \begin{equation}{ \varphi  }^{ -1 }=\frac { { \varphi  }^{\ast } }{ { f }_{ 0 }^{ 2 }\prod _{ k=1 }^{ n }{ \left( 1-\frac { { f }_{ k }^{ 2 } }{ { f }_{ 0 }^{ 2 } }  \right)  }  } , \end{equation} in case that only have scalar component  and the  ${ l }^{ ésima }$  direction component, the multiplicative inverse is written as follows \begin{equation}{ \varphi  }^{ -1 }=\frac { { \varphi  }^{ \ast } }{ \left( { f }_{ 0 }^{ 2 }-{ f }_{ l }^{ 2 } \right)  }. \end{equation}