[math-fun] quaternion identity
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
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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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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-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
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Column operations reduce the determinant to the form real times case a = 1, b = i, c = j, d = k, which in turn is easily seen to vanish. WFL On 7/18/12, Fred lunnon <fred.lunnon@gmail.com> 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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-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
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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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-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
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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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-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
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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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-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
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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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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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participants (6)
-
Dan Asimov -
Fred lunnon -
James Propp -
Mike Stay -
rcs＠xmission.com -
W. Edwin Clark