Quaternions

I've been struggling with this one for awhile, on and off for atleast 3 months. I've asked all my friends for help, and it seems Quaternions don't feel like working.

My main problem stems from this equation:

i^2 = j^2 = k^2 = ijk = − 1

No matter what, i,j,k are always -1..?

In a nut shell, I'd appreciate it if someone could give me a hand in solving Quaternions.

Thanks.

first of all, do you know what Quaternions are?  o.o   

In common usage, the Quaternions are a set of 2x2 matrices with complex entries, with this stipulation:  let H be the set of all quaternions.  (in the following entry, the double parentheses are a crappy way of representing 2x2 matrices).  Then,

H = { (a+bi   c+di) :   a,b,c,d are real numbers }
      (-c-di  a-bi)

where i is the imaginary unit.  Note that the bottom right element is the complex conjugate of the top left element, and the bottom left element is the negative complex conjugate of the top right element.  this is the defining characteristic of the quaternions.  now, what are i,j,k, and -1 for that matter?  They are defined as follows:

i= (i 0)    j= (0  1)   k= (0 i)   -1= (-1  0)
   (0 i),      (-1 0),     (i 0),      (0  -1)

note that these are all simple examples of elements in H.  Assuming you know matrix multiplication and multiplication of complex elements, you can go through and see that the four forms above do indeed turn out to be equal to the matrix -1.  Here's an example for i^2:

(i  0) . (i  0) = (-1  0) = -1. 
(0 -i)   (0 -i)   (0  -1)

Why is this so?  Because in terms of complex numbers, i x i = -1, and -i x -i = (-1)i x (-1)i = i x i = -1.  the other terms in the other parts of the dot products all cancel because you're just multiplying by zero of course.  in multiplying by i and -i, you have to mind your p's and q's.

Finally a bit of history.  The set is commonly denoted H after the discoverer of the quaternions, William Rowan Hamilton.  in a 'flash of genius', he scrawled the identity above into a Broom Bridge, somewhere in Ireland.  Hamilton became obsessed with quaternions later in life, trying to apply them to everything he could think of.  So quaternions are not like your average real numbers, nor are they just any old matrix.  to understand them, you'll have to understand 'complex conjugate', 'matrix multiplication', and operations on complex numbers in general.  I only mention this because your original problem with the identity seems to suggest that you are missing knowledge on at least one of these.  I could be wrong.  :)

Error:

in the description of H, the generic element "-c-di" should be "-c+di".

Hey that was pretty informative, for a world4ch post. In my CS class we only defined quaternions as (xi+yj+zk+w) with the usual properties of i,j,k. Nice to see how they're actually based on complex matrices.

Btw in your definition of i the lower row should read (0, -i).

Right you are.  Typo #2.

the (xi+yj+zk+w) of course is a generic linear combination of these basic elements of the quaternions.  If you've taken Calc 3, the same thing happens when we define the vectors i,j,and k in the sense of R3: EVERY vector in R3 can be represented as a linear combination of those three.  I think the term for a set of elements which generate every other element in a space via linear combination is "basis", but it's been a long time since linear algebra.

bump