Note the Aslaksen paper is dated 1996, not 1966 as stated in Wikipedia. Also my attempted proof that Mike Stay's 4x4 determinant vanishes is obviously rubbish: if column operations preserved the (Cayley) determinant, so would row operations, whence 2x2 and 3x3 determinants with equal rows would also vanish --- which they in general do not. WFL On 7/18/12, Fred lunnon <fred.lunnon@gmail.com> wrote:
The Aslaksen paper is available at http://arxiv.org/pdf/math/0111028
Does the complex-valued definition have the same value as the natural permutation-based definition in the case that either equals zero?
WFL
On 7/18/12, Dan Asimov <dasimov@earthlink.net> wrote:
It's interesting that in the Wikipedia article <http://en.wikipedia.org/wiki/Quaternionic_matrix>, it's stated that there is no natural way to define such a thing.
Much more detail is given in the fascinating article "Quaternionic determininants" by Aslaksen, referenced there, from the Math. Intelligencer, 1996.
I'm still not sure I believe there's no good quaternionic determinant, but especially the Aslaksen article makes a pretty good case for just that.
--Dan
On 2012-07-18, at 3:26 AM, Fred lunnon wrote:
This surely is just the 4x4 quaternion determinant with equal rows; but why exactly should that always vanish?
The phenomenon must somehow be connected with quaternions being 4-dimensional ...
WFL
On 7/18/12, Mike Stay <metaweta@gmail.com> wrote:
I think this is just the determinant of a 4x4 quaternion-valued matrix where each row is abcd. http://en.wikipedia.org/wiki/Levi-Civita_symbol
On Tue, Jul 17, 2012 at 6:19 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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