A basic reference on quaternionic determinants is Mehta, M. L. Random matrices. Third edition. Pure and Applied Mathematics (Amsterdam), 142. Elsevier/Academic Press, Amsterdam, 2004. Jim Propp On 7/18/12, W. Edwin Clark <wclark@mail.usf.edu> wrote:
If I am not mistaken this follows from the Amitsur-Levitzki Theorem which essentially says that the identity is satisfied by the algebra of 2x2 matrices over any commutative ring. Observing that the quaternions are isomorphic to a subalgebra of the 2x2 matrix algebra over the complex numbers we obtain the result for the quaternions. Clearly, any subring of a ring that satisfies an identity, satisfies the same identity. See: http://www.encyclopediaofmath.org/index.php/Amitsur%E2%80%93Levitzki_theorem and http://en.wikipedia.org/wiki/Quaternions#Matrix_representations
On Tue, Jul 17, 2012 at 9:19 PM, <rcs@xmission.com> wrote:
Has anyone seen this somewhere?
Quaternions a,b,c,d:
abcd+acdb+adbc+badc+bcad+bdca+**cabd+cbda+cdab+dacb+dbac+dcba = abdc+acbd+adcb+bacd+bcda+bdac+**cadb+cbad+cdba+dabc+dbca+dcab
The LHS is even permutations of abcd; RHS is odd.
It should also work for 2x2 matrices of real or complex numbers.
Rich
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