## Inverse

Here we look at the multiplicative inverse of multivector 'm', that is '1/m'. On this page we saw that the inverse of a pure vector is the same vector with a scaling factor.

#### Inverse of Bivector

On the same page we saw that the inverse of a bivector blade is calculated by replacing the bivector bases with their reverse (with a scaling factor). Taking the reverse of a bivector is the same as changing its sign. There is a problem in 4D and above though, the non-scalar terms may not cancel out because bivectors like e12 and e34 don't have a common term. So do four dimensional bivectors always have a multiplicative inverse?

Note on terminology: its not a good idea to use the division symbol '/' or '÷' because order is important for multivector multiplication, so when we write: a/b, is not clear if we mean a*(b^{-1}) or (b^{-1})*a. However I will use the '/' symbol here with the understanding that it means: a/b = a*(b^{-1}).

Lets try inverting the bivector: a e12 + b e34, this multiplying top & bottom by the reverse gives:

So the denominator still has a non-scalar term, we can make another attempt to cancel it out by multiplying by its reverse, so the numerator is,

(a e_{21} + b e_{43})(a² + b² + 2ab e_{1234})

= (a (a² + b²) - 2ab²) e_{21} + (b (a² + b²) - 2a²b) e_{43}

= (a (a² - b²)) e_{21} + (b (b² - a²)) e_{43}

= (a (a + b)(a - b)) e_{21} + (b (b - a)(b + a)) e_{43}

and the denominator is:

(a² + b² - 2ab e_{1234})(a² + b² + 2ab e_{1234})

= (a + b)²(a² + b² - 2 a b)

so the result is:

= (a (a - b)/((a + b)(a² + b² - 2 a b))) e_{21}

+ (b (b - a)/((a + b)(a² + b² - 2 a b))) e_{43}

This looks very messy. Can it be simplified and further?

So how can we calculate the inverse for a general bivector based in 4D which contains up to 6 terms? The inverse of any two terms with a common term is the reverse, but if there is no common term the result is more complicated as we have seen.

The general case for the inverse of a 4D bivector:

1/ (a e12 + b e31 + c e23 + d e41 + f e42 + g e43)

is:

(-a e12 - b e31 - c e23 - d e41 - f e42 - g e43)(x - y e1234)/(x² - y²)

= ((g y -a x) e12

+ (f y - b x) e31

+ (d y - c x) e23

+ (c y - d x) e41

+ (b y - f x) e42

+ (a y - g x) e43)/(x² - y²)

where:

- x = a² + b² + c² + d² + f² + g²
- y =2*(a*g + b*f + c*d)

#### Inverse of 4D Trivector

1/(a e_{123} + b e_{142} + c e_{134} + d e_{324})

is equal to

(-a e_{123} - b e_{142} - c e_{134} - d e_{324})/(a² + b² + c² + d²)

So we don't have the same problem for trivectors that we had for bivectors, in 4d we can just use the reverse of all the terms, with the appropriate scaling factor.

#### Inverse of 4D Quadvector

1/(a e_{1234}) = (1/a) e_{1234}

There is only one quadvector in 4D which is its own inverse.

## Dual

This multiplies by e_{1234} , for general information about n-dimensional dual see this page.

This is the bottom row of the multiplication table here.

A _{r} |
dual(A_{r}) = A_{r}^{*} = e_{1234} A_{r} |

e | e_{1234} |

e_{1} |
e_{1234} e_{1}= e_{324} |

e_{2} |
e_{1234} e_{2}= e_{134} |

e_{3} |
e_{1234} e_{3}= e_{142} |

e_{4} |
e_{1234 }e_{4}= e_{123} |

e_{12} |
e_{1234} e_{12}= e_{43} |

e_{31} |
e_{1234} e_{31}= e_{42} |

e_{23} |
e_{1234} e_{23}= e_{41} |

e_{41} |
e_{1234} e_{41}= e_{23} |

e_{42} |
e_{1234} e_{42}= e_{31} |

e_{43} |
e_{1234} e_{43}= e_{12} |

e_{123} |
e_{1234} e_{123}= e_{4} |

e_{142} |
e_{1234} e_{142}= e_{3} |

e_{134} |
e_{1234} e_{134}= e_{2} |

e_{324} |
e_{1234} e_{324}= e_{1} |

e_{1234} |
e_{1234} e_{1234}= e _{} |

## Reverse

The reverse function of a multivector reverses the order of its factors, including the order of the base values within a component. The reverse function is denoted by †, so the reversal of A is denoted by A†.

For general information about n-dimensional reverse see this page.

A _{r} |
A_{r}† |

e | 1 |

e_{1} |
e_{1} |

e_{2} |
e_{2} |

e_{3} |
e_{3} |

e_{4} |
e_{4} |

e_{12} |
e_{21} = -e_{12} |

e_{31} |
e_{41} = -e_{31} |

e_{23} |
e_{32} = -e_{23} |

e_{41} |
e_{24}= -e_{41} |

e_{42} |
e_{32}= -e_{42} |

e_{43} |
e_{34}= -e_{43} |

e_{123} |
e_{321}= -e_{123} |

e_{142} |
e_{241} = -e_{142} |

e_{134} |
e_{431} = -e_{134} |

e_{324} |
e_{423} = -e_{324} |

e_{1234} |
e_{4321}= e_{1234} |

The reversal function is important for a number of reasons, one reason is that it can map a multiplication into another multiplication with the order of the multiplicands reversed:

(A * B)† = B†* A†

We can think of this as a morphism where † maps to an equivalent expression with order of multiplication reversed.

Another application for the reversal function is to specify a transformation from one vector field to another:

p_{out} = A p_{in} A†

In other words, if p_{in} is a pure vector (i.e. real, bivector and tri-vector parts are all zero) then p_{out} will also be a pure vector. I would appreciate any proof of this.If you have a proof, that I could add to this page, please let me know, In the 3D case I guess I could try the brute force approach of multiplying out the terms for a general expression A and p.

I think this may also apply to:

p_{out} = A p_{in} A ^{-1}

but not every multivector is invertible, one condition that should ensure that a multivector is invertible is:

A A† = 1

## Conjugate

The conjugate of A_{r} is denoted A_{r}^{~} where: A_{r}^{~}*A_{r} = I = psudoscalar, for general information about n-dimensional conjugate see this page.

If A_{r} represents a transformation then A_{r}^{~} reverses the transformation

A _{r} |
A_{r}^{~}=(A_{r}†)^{*} |
A_{r}^{~}*A_{r} |

e | (1)^{*} = e_{1234} |
e_{1234} |

e_{1} |
(e_{1})^{*} = -e_{234} |
e_{1234} |

e_{2} |
(e_{2} )^{*} = e_{431} |
e_{1234} |

e_{3} |
(e_{3} )^{*} = -e_{124} |
e_{1234} |

e_{4} |
(e_{4} )^{*} = -e_{321} |
e_{1234} |

e_{12} |
(-e_{12} )^{*} = e_{34} |
e_{1234} |

e_{31} |
(-e_{31} )^{*} = e_{23} |
e_{1234} |

e_{23} |
(-e_{23} )^{*} = e_{42} |
e_{1234} |

e_{41} |
(-e_{41} )^{*} = e_{13} |
e_{1234} |

e_{42} |
( -e_{42})* = e_{14} |
e_{1234} |

e_{43} |
( -e_{43} )^{*} = e_{12} |
e_{1234} |

e_{123} |
( -e_{123} )^{*} = -e_{4} |
e_{1234} |

e_{142} |
( -e_{142} )^{*} = e_{3} |
e_{1234} |

e_{134} |
( -e_{134} )^{*} = e_{2} |
e_{1234} |

e_{324} |
( -e_{324} )^{*} = e_{1} |
e_{1234} |

e_{1234} |
( e_{1234} )^{*} = 1 |
e_{1234} |

The last column confirms that A_{r}^{~}*A_{r} = I = pseudoscalar.

## Norm

under construction

## Exponential

We define the exponential of multivector a as ea º åi=0¥ ai
/ i ! .

In particular:

ea is the traditional scalar exponential function

For any pure square multivector [ a2 = ±|a|2 ] we have ea = cos|a| + a~
sin|a| if a2 < 0 ;

cosh|a| + a~ sinh|a| if a2 > 0 ;

1+a if a2 = 0.

For unit multivector a:

eafeay = ea(f+y)

(d/df) elaf =lea(lf+p/2) =laelaf

## Involution

a# = sum _{k=0}(^{N} (-1)^{k} a _{<k>} = a_{<+>} - a_{<->} .