" tommy quaternion "

[tommy1729]

Can you post the matrix representation?

Hi marraco.
These numbers have NO matrix representation because multiplication is NOT associative.
So doing simple linear algebra probably will not work.

This is one of the simplest algebra's that have no matrix representation.
Or should I say smallest.
Dimension 3 does not have this property if we require it to have some group like structure.
So dimension 4 is the smallest.

HOWEVER it is not the only one in dimension 4.
There exists at least one other nonisomorphic one :


unfortunately neither of them are closed under square root.

( \( 1 + 2 A + 3 B + 4 C \)  has no square root in either one )



Another one is 

\( A*B=B*A=C,A*C=C*A=B \)

\( B*C=C*B=A \)

\( A*A=B*B=-1 \)

\( C*C=1 \)

Thank you for your interest.


regards

tommy1729