Guess it's been too long since I thought about quaternions. Multiplicative
inverse is commutative, but multiplication in general is not. I guess I had
been thinking about matrix inverse in quaternions. When I try various
methods to get the inverse they don't work. That was a while back when I was
trying that.
Tried the Tailor expansion for 1 o. 1i2j3k4 and it checked out.
t=.(1&o.) t. i.30
Q''
Entering Quaternion Mode
+/t*1i2j3k4^i.30
91.7837i21.8865j32.8297k43.773
(1 o. 1i2j3k4)-+/t*1i2j3k4^i.30
_5.47118e_12i3.05533e_12j4.5759e_12k6.11067e_12
On Sun, Apr 27, 2008 at 11:31 PM, Kip Murray <[EMAIL PROTECTED]> wrote:
> About multiplicative inverses, Birkhoff and Mac Lane's A Survey of
> Modern Algebra says quaternions have them. Their explanation: if
>
> x = a + b i + c j + d k (conventional notation, a b c d are real)
>
> and you define
>
> x* = a - b i - c j - d k
>
> then
>
> x x* = x* x = a^2 + b^2 + c^2 + d^2
>
> so as long as one of a b c d is not zero, x has the inverse
>
> x*/(a^2 + b^2 + c^2 + d^2) .
>
>
> (It is easy to see that
>
> x (inverse of x) = (inverse of x) x = 1 .)
>
>
> Kip Murray
>
>
> On Sun, 27 Apr 2008, Don Guinn wrote:
>
> Well, the problem with quaternions is that multiplication is not
> commutative. Therefore there is no such thing as a multiplicative
> inverse.
> You can have a left inverse or a right inverse. Fortunately, they are
> conjugates, depending on how you define conjugate. In calculating
> transcendentals, computers typically pick a "principle" solution. There
> are
> many solutions to something like the ArcSin of x. Where is the principle
> solution in quaternions?
>
> Take "a*b" is equivalent to "a(+&.^.)b" in real and complex numbers. Not
> necessarily in quaternions. Picking principle solutions in the typical
> way
> does not work. But, interestingly, there are solutions for ^. in
> quaternions
> where that relationship is true.
>
> I defined the sin(x) as follows. Right now I don't remember where I found
> that definition. Long time ago. I slept since then.
>
> Qsin=: 3 : 0"0
> 't V mV'=. QtVm y
> Qcloseq((sin t)*(cosh mV)),((cos t)*(sinh mV)*(V%mV))
> )
>
> where
>
> QtVm=: (({.;}.),[:<[:+/&.:*:}.)&>
> NB.*QtVm - Break a quaternion into real, imaginary and magnitude.
> NB. Parts consist of t,V and |y|
>
> and
>
> sin=: 1&o.
> cos=: 2&o.
> sinh=: 5&o.
> cosh=: 6&o.
>
> Say something like
>
> 1 o. 1i2j3k4
> 91.7837i21.8865j32.8297k43.773
>
> Is this a good principle value for 1i2j3k4? I'm not sure.
>
> It would be interesting to see if a Taylor series gave the same answer. I
> haven't tried that yet. Will have to.
> On Sun, Apr 27, 2008 at 7:43 PM, Raul Miller <[EMAIL PROTECTED]>
> wrote:
>
> > On Sun, Apr 27, 2008 at 8:42 AM, Don Guinn <[EMAIL PROTECTED]> wrote:
> > > I took a shot at defining the circular functions in quaternions. I'm
> not
> > > sure that they are correct.
> >
> > Taylor Series?
> >
> > --
> > Raul
> > ----------------------------------------------------------------------
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> >
> >
> ----------------------------------------------------------------------
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>
>
> Kip Murray
>
> [EMAIL PROTECTED]
> http://www.math.uh.edu/~km
> ----------------------------------------------------------------------
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>
>
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