Maths - Multivector Powers

We want to calculate an expression for:

(m)n

where:

When we calculated the powers of vectors we saw that, if the vectors anti-commute (as they do in euclidean space) then the result was not necessarily closed within the vector space, if 'n' is even then the result will be a pure scalar value and if 'n' is odd then the result will be a vector. So:

grade n result
vector even scalar
vector odd vector

If we now go on to try this for pure bivectors we can see that:

grade n result
bivector even scalar
bivector odd bivector

So we can see that when m is any pure grade then:

So what if m is any mixed grade multivector?

Square

Lets start by assuming that the multivector squares to a scalar value 's' that is: (m)²=s this assumption only applies when the real part of the multivector is zero. This value may be positive or negative since if vectors square to +ve then bivectors square to -ve so depending on which is greater will determine the sign of the square.

Other powers

So even powers will have scalar values, here are the first few powers:

power value type
s scalar (+ve or -ve)
m*s multivector
m4 scalar (+ve)
m5 m*s² multivector

I cant think of a way to analyse the more general case of a non-zero real part, I've tried this with the specific case of 2D vectors here but things get too messy.


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Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

flag flag flag flag flag flag Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics (Fundamental Theories of Physics). This book is intended for mathematicians and physicists rather than programmers, it is very theoretical. It covers the algebra and calculus of multivectors of any dimension and is not specific to 3D modelling.

 

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