From: Robin Chapman <rjchapman@my-deja.com>
Subject: Re: Scalar matrix polynomial
Date: Thu, 04 Jan 2001 10:07:39 GMT
Newsgroups: sci.math
Summary: Amitsur-Levitzki theorem (matrix rings are Polynomial-Identity rings)

In article <931c4f$q2g$1@nnrp1.deja.com>,
  ben_geffen@my-deja.com wrote:
> If A and B are 2 by 2 matrices over a field K then the matrix
> C = A*B-B*A has zero trace and therefor C*C is a scalar matrix (by the
> Cayley Hamilton theorem for example) so the polynomial:
> (A*B-B*A)^2 takes only scalar matrix values for any A and B.
>
> Is there nontrivial polynomial that its values are always scalar
> matrices also for matrices of size 3 by 3 ?

Define, for positive integer k,
F_k(A_1, ..., A_k)
to be the sum of sign(sigma) A_{sigma(1)}...A_{sigma(k)}
where sigma ranges over the permutations of {1,2,...,k}.
For example F_3(A,B,C) = ABC + BCA + CAB - ACB - BAC - CBA.

The Amitsur-Levitzki theorem states that
F_{2n}(A_1,...,A_{2n}) = 0 whenever A_1,..., A_{2n}
are n by n matrices over a commutative ring.

--
Robin Chapman, http://www.maths.ex.ac.uk/~rjc/rjc.html

 "His mind has been corrupted by colours, sounds and shapes."
   The League of Gentlemen


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==============================================================================

From: tim_robinson@my-deja.com
Subject: Re: Scalar matrix polynomial
Date: Thu, 04 Jan 2001 11:30:48 GMT
Newsgroups: sci.math

In article <931i19$u55$1@nnrp1.deja.com>,
  Robin Chapman <rjchapman@my-deja.com> wrote:

[quote of most of previous message deleted --djr]

> For example F_3(A,B,C) = ABC + BCA + CAB - ACB - BAC - CBA.
>

Aha!  This looks like a determinant of some sort.

Something like:

| A A A |
| B B B |
| C C C |

This is an abuse of the notation, but it shows very clearly what is
going on.



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