Quaternions : n-Space Basis Matrices

Hello, 3rd Apr, 2005, ~ 5.45 PM

I experimented with Quaternions by trying out with various values of the coeffs a, b, c, d.

For eg, I experimented with Int Coeffs, Real Coeffs and also Complex Coeffs.

It will be good while considering the Questions below, to associate them with the Observations indicated below.

TIA and Rgds

Sundar Krishnan

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I have referred to the foll MW refs in the discussions below.
http://mathworld.wolfram.com/Quaternion.html http://mathworld.wolfram.com/notebooks/Algebra/Quaternion.nb

MW gives 2 formulae for Quaternions :
one in R2 Space, and one in R4 Space.
[2 in R2, and 4 in R4, are to be read as Superscripts on R.]

In both cases, we have :

H = a*1 + b*i + c*j + d*k

conj_H = a*1 - b*i - c*j - d*k

but the Basis Matrices are defined differently - see (2) below.

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Questions :

(1) When do we use R2 and when R4 ?
Some non-trivial, real-life examples for both cases will be preferred.

Pl also note in (4) below, the diff in behaviour when we have Complex Coeffs.
Are there any real-life examples with Complex Coeffs ?
If yes, it would be good to understand the diff in behaviour between R2 and R4 Spaces in the light of your citing examples.

(2) Can we extend to Rn space ? How does one go about in arriving at the Basis Matrices for n-space ? See (2) below.

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Some observations :

(1)
R2 formulae gives Complex 2 x 2 Matrices - let us call this matrix H_Q8_R2.
R4 formulae gives 4 x 4 Real Matrices - let us call this matrix H_Q8_R4.

(2)
There are 2 sets of Basis Matrices :

The ones used in the formulae for R2 are 2 x 2 Complex Matrices – called U, I, J, K.
The ones used in the formulae for R4 are 4 x 4 Real Matrices – called i, j, k, 1.

(3)
When Coeffs a, b, c, d are Real, the Conjugate Formulae actually yields the "Conjugate Transpose" of H_Q8_R2, not True Transpose.
[For those who understand Matlab notation, "." (dot) is Conj Transp ; .' (dot, dash) is True Transpose.]

(4) Complex Coeffs a, b, c, d :
Since we finally need 2 Complex nos, I experimented in this case with :
Cab = a + b ; Ccd = c + d

The observations are :

a) In R2 Space :
If H_Q8_R2 forms a 2 x 2 pattern : [ P, Q ; R, S ], then the Conjugate Formulae yields a 2 x 2 matrix of pattern : [S, -Q ; -R, P].

b) In R4 Space :
Now, for Cab and Ccd, H_Q8_R4 is Complex, and the Conjugate Formulae yields True Transpose [eqvt to .' (dot, dash)].

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