Quaternions

[Kyon]

Complexity to the fourth dimension.

So, I just got into this subject that mathematicians refer to as "quaternions". And, since this is the lounge sub-forum, I thought I should ask a few questions about it..
When I have two quaternions, Q₁ and Q₂, respectively assigned the following values for their real components (in a+bi+cj+dk, a, b, c and d are the real components): {2, 3, 4, 5} and {3, 4, 5, 6}.
When I then type this into my calculator and do some simple matrix calculations:
Q₁*Q₂ = {68}
Q₂*Q₁ = { {6, 9, 12, 15}, {8, 12, 16, 20}, {10, 15, 20, 25}, {12, 18, 24, 30} }
(Note: I defined Q₁ as a 1x4 Matrix and Q₂ as a 4x1 Matrix.)

I understand the second output, since this is just 3 * {2, 3, 4, 5}, 4 * {2, 3, 4, 5}, etc. But, what does the first calculation evaluate to, what does this number represent?

More questions might pop-up on the way, as this is a rather new subject to me.

Thanks in advance for even reading this topic up to this point.


EDIT:
Changed the Q₁*Q₂ equation, was not paying attention when writing it down.

[m4ster r0shi]

Mmmm... What you do there is matrix multiplication, not quaternion multiplication. You are almost right about Q₂*Q₁ (it's 3*{2,3,4,5}, 4*{2,3,4,5}, ...). As for Q₁*Q₂, the result is calculated like this:

2*3 + 3*4 + 4*5 + 5*6 = 68 (I use bold for the Q₁ elements).

Although I have a maths degree, quaternions wasn't a mandatory course, thus I'm not familiar with the subject (but I'm quite sure that this is not how you multiply quaternions). All I know is that they are used in 3D graphics for rotation. Maybe hamsterman (or someone else) can explain this better.

[Kyon]

Whoops, was not paying attention when writing down the equations, too lazy to change them now, too. So, yeah, that would make sense. This does have a lot to do with quaternion, since they are expressed in 4 "dimensions" (1 real and 3 imaginary). A quaternion is a "more complex" number than regular imaginary numbers:

i² = j² = k² = ijk = -1

We can now get the value of i, j, and k, expressed in the other variables like this:
-1 = ijk
Times i
-i = i²jk
-i = (-1)jk
i = jk

Doing the same for the others, we get:
i = jk
j = ik
k = ij
-i = kj
-j = ki
-k = ji

And there's a bunch of other things going on which I can't explain better than a simple google search.

EDIT:
Just another thing, the latter equation has something to do with quaternions, as it returns a set of multiplications of 1, i, j and k, but.. what does the other value represent in this context?

EDIT:
Found out the striped through question; the values are irrelevant, that's all there is to say about it.

[tition]

Umm, one natural way to write down quaternions is with 4x4 matrices, ( I doubt you'll get something good with 1x4 matrices).

Wikipedia tells you how to write quaternions with 2x2 *complex* matrices. You can turn any n by n complex matrix to a real 2n by 2n matrix by substituting
a+ib
with the matrix

(a -b)
(b  a)

So, the 4 complex matrices (2 by 2 each) given in Wikipedia can be written as 4x4 real matrices:

1->

(1  0  0  0)
(0  1  0  0)
(0  0  1  0)
(0  0  0  1)

i->

(0 -1  0  0)
(1  0  0  0)
(0  0  0  1)
(0  0 -1  0)

j->

(0  0  1  0)
(0  0  0  1)
(-1 0  0  0)
(0 -1  0  0)

k->

(0  0  0 -1)
(0  0  1  0)
(0 -1  0  0)
(1  0  0  0)

You can now verify that the defining equalities you wrote for the quaternions hold for these 4x4 matrices.

Taking linear combinations of them gets you all quaternions.

A nice exercise: write a parser for quaternion expressions. Then simplify the quaternion expressions. Finally, print out the matrix form of the result.

Of coures, such an exercise would take you looooots of time.

[cloudsINmybrain]

kyon, it is an interesting area of math.

I find thinking of it as "directionally determined " multiplication through the standard i - j - k orientation is helpful. When you go the opposite direction you get the negative. In the above you have:

j = ik and -j = ki

That looks reversed to me and should be j = ki and -j = ik.

[Kyon]

That looks reversed to me and should be j = ki and -j = ik.

ijk = -1
i(j²)k = -j
i(-1)k = -j
ik = j

[Kyon]

tition wrote:

1->

(1  0  0  0)
(0  1  0  0)
(0  0  1  0)
(0  0  0  1)

Wouldn't that be:

(1  0  0  0)
(0 -1  0  0)
(0  0 -1  0)
(0  0  0 -1)

, considering that i² = j² = k² = -1?
Since all of them are just rewrites of the multiplication tables (with the signs of the relevant values).

[tition]

Since all of them are just rewrites of the multiplication tables (with the signs of the relevant values).

Nope, they aren't.

You are right, it is a rewrite of the multiplication table. As 1*i=i, 1*j=j, 1*k=k, the signs should be +1.

P.S. sorry for the late reply.

[Kyon]

Just, how is a quaternion multiplication performed, given two simple example quaternions?