Maths - Maths - 5D Multivector basis - Martin Baker

Maths - choosing bases - 5D

The bases are the vectors e1, e2, e3, e4 and e5, however the higher level products such as: e1^e2, e3^e1, e2^e3, e1^e4, e4^e2 and e3^e4 is also as a basis for the bivectors.

The question is what order do we choose for this (which is equivalent to saying what sign do we use since e1^e2=-e2^e1).

The methodology used for choosing the order of these indexes is explained here.

We calculate the psuedoscalar by drawing the matrix formed by putting the basis vectors side-by-side, then taking its minor by removing the row associated with its own coordinate type and removing the column of the basis vector not associated with.

We have to be very careful with signs as the sign alternates with terms as follows:

e1x	e2x	e3x	e4x	e5x
e1y	e2y	e3y	e4y	e5y
e1z	e2z	e3z	e4z	e5z
e1w	e2w	e3w	e4w	e5w
e1v	e2v	e3v	e4v	e5v

=e1x*

e2y	e3y	e4y	e5y
e2z	e3z	e4z	e5z
e2w	e3w	e4w	e5w
e2v	e3v	e4v	e5v

-e2x*

e1y	e3y	e4y	e5y
e1z	e3z	e4z	e5z
e1w	e3w	e4w	e5w
e1v	e3v	e4v	e5v

+e3x*

e1y	e2y	e4y	e5y
e1z	e2z	e4z	e5z
e1w	e2w	e4w	e5w
e1v	e2v	e4v	e5v

-e4x*

e1y	e2y	e3y	e5y
e1z	e2z	e3z	e5z
e1w	e2w	e3w	e5w
e1v	e2v	e3v	e5v

+e5x*

e1y	e2y	e3y	e4y
e1z	e2z	e3z	e4z
e1w	e2w	e3w	e4w
e1v	e2v	e3v	e4v

To choose tri-vectors for 5D multivectors we start with the psudoscalar e1^e2^e3^e4^e5 which represents the whole determinant. We then split it up, taking into account the sign, as above:

e1^e2^e3^e4^e5 = e1 * e2^e3^e4^e5 - e2 * e1^e3^e4^e5 + e3 * e1^e2^e4^e5 - e4 * e1^e2^e3^e5 + e5 * e1^e2^e3^e4

Where the sign is negative then we invert the order, which gives the basis tri-vectors as follows:

e2^e3^e4^e5 is the minor of e1
-e1^e3^e4^e5 is the minor of e2
e1^e2^e4^e5 is the minor of e3
-e1^e2^e3^e5 is the minor of e4
e1^e2^e3^e4 is the minor of e5

In order to remove the minus terms from the 2nd and 4th terms we can just swap any two of the terms, but which two do we swap?

how about:

e2^e3^e4^e5 is the minor of e1
e3^e1^e4^e5 is the minor of e2
e1^e2^e4^e5 is the minor of e3
e1^e2^e5^e3 is the minor of e4
e1^e2^e3^e4 is the minor of e5

I'm not an expert at suduko, so I cant get one number in each row, but the above seems to have a patern. Can anyone tell me a better way to choose the order?

What are the properties of these under the dual function?

So this is the best I can do to get the order for the quad-vectors.

What about the bivectors and tri-vectors? we can reduce the deteminant in two stages, but how does that correspond to the clifford algebra functions?

I think I will try finding the minors of the above tri-vectors as follows:

The minor of e1

The minor of e2

The minor of e3

The minor of e4

The minor of e5

e2^e3^e4^e5

e1^e3^e4^e5

e1^e2^e4^e5

e1^e2^e3^e5

e1^e2^e3^e4

Now take the minor of e1:

which gives: e3^e4^e5

Now take the minor of e1:

which gives: -e2^e4^e5

Now take the minor of e1:

which gives: e2^e3^e5

Now take the minor of e1:

which gives: -e2^e3^e4

Now take the minor of e2:

which gives: e3^e4^e5

Or take the minor of e2:

which gives: e1^e4^e5

Or take the minor of e2:

which gives: -e1^e3^e5

Or take the minor of e2:

which gives: e1^e3^e4

Or take the minor of e3:

which gives: -e2^e4^e5

Or take the minor of e3:

which gives: e1^e4^e5

Or take the minor of e3:

which gives: e1^e2^e5

Or take the minor of e3:

which gives: -e1^e2^e4

Or take the minor of e4:

which gives: e2^e3^e5

Or take the minor of e4:

which gives: -e1^e3^e5

Or take the minor of e4:

which gives: e1^e2^e5

Or take the minor of e4:

which gives: e1^e2^e3

Or take the minor of e5:

which gives: -e2^e3^e4

Or take the minor of e5:

which gives: e1^e3^e4

Or take the minor of e4:

which gives: -e1^e2^e4

Or take the minor of e5:

which gives: e1^e2^e3

so to summarise:

minor of e1 followed by e2 gives:	e1^e2	e3^e4^e5	(e1^e2)(e3^e4^e5)
minor of e1 followed by e3 gives:	e1^e3	-e2^e4^e5	-(e1^e3)(e2^e4^e5)
minor of e1 followed by e4 gives:	e1^e4	e2^e3^e5	(e1^e4)(e2^e3^e5)
minor of e1 followed by e5 gives:	e1^e5	-e2^e3^e4	-(e1^e5)(e2^e3^e4)
minor of e2 followed by e1 gives:	e2^e1	-e3^e4^e5	-(e2^e1)(e3^e4^e5)
minor of e2 followed by e3 gives:	e2^e3	e1^e4^e5	(e2^e3)(e1^e4^e5)
minor of e2 followed by e4 gives:	e2^e4	-e1^e3^e5	-(e2^e4)(e1^e3^e5)
minor of e2 followed by e5 gives:	e2^e5	e1^e3^e4	(e2^e5)(e1^e3^e4)
minor of e3 followed by e1 gives:	e3^e1	e2^e4^e5	(e3^e1)(e2^e4^e5)
minor of e3 followed by e2 gives:	e3^e2	-e1^e4^e5	-(e3^e2)(e1^e4^e5)
minor of e3 followed by e4 gives:	e3^e4	e1^e2^e5	(e3^e4)(e1^e2^e5)
minor of e3 followed by e5 gives:	e3^e5	-e1^e2^e4	-(e3^e5)(e1^e2^e4)
minor of e4 followed by e1 gives:	e4^e1	-e2^e3^e5	-(e4^e1)(e2^e3^e5)
minor of e4 followed by e2 gives:	e4^e2	e1^e3^e5	(e4^e2)(e1^e3^e5)
minor of e4 followed by e3 gives:	e4^e3	-e1^e2^e5	-(e4^e3)(e1^e2^e5)
minor of e4 followed by e5 gives:	e4^e5	e1^e2^e3	(e4^e5)(e1^e2^e3)
minor of e5 followed by e1 gives:	e5^e1	e2^e3^e4	(e5^e1)(e2^e3^e4)
minor of e5 followed by e2 gives:	e5^e2	-e1^e3^e4	-(e5^e2)(e1^e3^e4)
minor of e5 followed by e3 gives:	e5^e3	e1^e2^e4	(e5^e3)(e1^e2^e4)
minor of e5 followed by e4 gives:	e5^e4	-e1^e2^e3	-(e5^e4)(e1^e2^e3)

So the full multivector and its dual is:

A_r	dual(A_r) = A_r^* = e₁₂₃₄₅ A_r
1	e₁₂₃₄₅
e₁	e₁₂₃₄₅e₁= e₂₃₄₅
e₂	e₁₂₃₄₅e₂= e₃₁₄₅
e₃	e₁₂₃₄₅e₃= e₁₂₄₅
e₄	e₁₂₃₄₅e₄ = e₁₂₅₃
e₅	e₁₂₃₄₅e₅ = e₁₂₃₄
e₁₂	e₁₂₃₄₅e₁₂= -e₃₄₅
e₁₃	e₁₂₃₄₅e₁₃= -e₄₂₅
e₁₄	e₁₂₃₄₅e₁₄= -e₂₃₅
e₁₅	e₁₂₃₄₅e₁₅= -e₃₂₄
e₂₃	e₁₂₃₄₅e₂₃= -e₁₄₅
e₂₄	e₁₂₃₄₅e₂₄= -e₃₁₅
e₂₅	e₁₂₃₄₅e₂₅= -e₁₃₄
e₃₄	e₁₂₃₄₅e₃₄= -e₁₂₅
e₃₅	e₁₂₃₄₅e₃₅= -e₂₁₄
e₄₅	e₁₂₃₄₅e₄₅=-e₁₂₃
e₁₂₃	e₁₂₃₄₅e₁₂₃= -e₄₅
e₂₁₄	e₁₂₃₄₅e₂₁₄= -e₃₅
e₁₂₅	e₁₂₃₄₅e₁₂₅= -e₃₄
e₁₃₄	e₁₂₃₄₅e₁₃₄= -e₂₅
e₃₁₅	e₁₂₃₄₅e₃₁₅= -e₂₄
e₁₄₅	e₁₂₃₄₅e₁₄₅= -e₂₃
e₃₂₄	e₁₂₃₄₅e₃₂₄= -e₁₅
e₂₃₅	e₁₂₃₄₅e₂₃₅= -e₁₄
e₄₂₅	e₁₂₃₄₅e₄₂₅= -e₁₃
e₃₄₅	e₁₂₃₄₅e₃₄₅=-e₁₂
e₁₂₃₄	e₁₂₃₄₅e₁₂₃₄= e₅
e₁₂₅₃	e₁₂₃₄₅e₁₂₅₃= e₄
e₁₂₄₅	e₁₂₃₄₅e₁₂₄₅= e₃
e₃₁₄₅	e₁₂₃₄₅e₃₁₄₅= e₂
e₂₃₄₅	e₁₂₃₄₅e₂₃₄₅= e₁
e₁₂₃₄₅	e₁₂₃₄₅e₁₂₃₄₅= 1

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see also:

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Maths - choosing bases - 5D

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