### Quaternion or quaternion operator

greenspun.com : LUSENET : quaternions : One Thread

I arrive at (a-i*a)^2/(1-i)/a^2 = 1. The implication is that if a is some kind of operator. It seems quat-ish. Is it?

There's an other solution that I cannot derive but sense that is in the matrix form [-1-Ai, 1-Ai; A+i, -A+I]

Does either make sense?

-- stephen nagy (stephennagy@mindspring.com), April 07, 2002

Hello Stephen:

The short answer is that I just do not understand your terminology, so the question makes no sense to me.

Most of the quaternion operators I use are defined on the manifold of R4, meaning 4 real numbers. The differential quaternion operator on R4 looks like this: (d/dt, d/dx, d/dy, d/dz). To make things a little more compact, I sometimes will us a capital letter to symbolize the 3-vector, so in the previous example, I might write as (d/dt, Del). This operator then gets multiplies a function, (f, F), generating:

(d/dt, Del)(f, F) = (df/dt - Del.F, dF/dt + Del f + DelxF) time div time grad curl der. der scalar vector

This appears to my eye as a complete assessment of how a 4D function can change, one of the strengths of working with quaternions in spacetime, a 4D real manifold of importance.

Real quaternions can always be written as a matrix, even the above example. Quaternions are consistent :-)

|t -x -y -z| q(t,x,y,z) = |x t -z y| |y z t -x| |z -y x t|

When working with a symbolic math package such as Mathematica, I almost always use this representation as it is reliable. The complex unit vector i confuses such programs and even humans. Remember, there is an i, a j and a k to deal with that do not commute with each other.

One could try to work with the one dimensional manifold of H^1. This may become a very important area of study for quaternion analysis, but I will need help from professional mathematicians in this area.

Good luck in your work, doug

-- Douglas Sweetser (sweetser@theworld.com), April 08, 2002.

Hello Doug

Is it possible to send an attachment file with eg. extension .rtf to you? I have matrix represantions, which should require this

Pekka

-- Pekka Tapio Laakso (pekkat.laakso@mail.suomi.net), January 27, 2003.