Re: Quaternions : n-Space Basis Matrices

Hello, 3rd Apr, 2005, ~ 10 PM

Sorry, there is a correction in the last message posted wrt the foll section on Observations wrt Complex Coeffs :

Pl replace that particular section : (4) under Observations with the foll.
The complete revised section (4) pertaining to Complex Coeffs is given below.

Inconvenience regretted.

Sundar Krishnan

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Some observations :
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(4) Complex Coeffs a, b, c, d :

Since we finally need 2 Complex nos, I experimented in this case with :
Cab = a + b *j ; Ccd = c + d *j [forgot "*j" earlier]

The observations when using Cab and Ccd are :

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

b) In R2 Space :
Eqn (3) in http://mathworld.wolfram.com/Quaternion.html gives an expr for H.
Let us call it H_Q8_R2_Cabcd. This expression does not tally with H_Q8_R2 - obviously because H_Q8_R2 considers the Complex Coeffs in raw input form.

c) In R4 Space :
Now, (note : with Complex a, b, c, d), H_Q8_R4 is a Complex Matrix, and the Conjugate Formula yields True Transpose [eqvt to .’ (dot, dash)].

d) In R4 Space :
Is there an eqvt formula in R4 Space - let us call it say, H_Q8_R4_Cabcd - similar to Eqn (3) for R2 Space : H_Q8_R2_Cabcd ?

In other words, do we have an alternate formula for R4 Space in matrix form similar to Eqn 3 for R2 Space ? If so, the matrix H_Q8_R4 may coincide with H_Q8_R4_Cabcd atleast for Real Coeffs ??

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