The Amitsur-Levitzki Theorem for a Non Mathematician.

Yaacov Levitzki

The purpose of this post is to describe the Amitsur-Levitzki theorem: It is meant for people who are not necessarily mathematicians. Yet they need to know two things. The first is what matrices are. Very briefly, matrices are rectangular arrays of numbers. The second is that two n by n square matrices A and B can be multiplied. We denote their product by A x B and the product is again an n by n matrix. Unlike numbers, the product of matrices need not be commutative.  In other words A x B can be different from B x A.  The Wikipedia article about matrices is a good source.

We can multiply more than two matrices. We can write A x B x C for the product of A, B and C. The order of the matrices is important, but the order in which we perform the multiplication is not. This is because multiplication of matrices is associative,  that is

 (A x B ) x C  = A x (B x C).

Here is the Amitsur Levitzki Theorem for 2 x2 matrices:

For every four 2 x 2 matrices A, B, C, and D

A x B x C x D – B x A x C x D – A x B x D x C + B x A x D x C – A x C x B x D + C x A x B x D  +  A x C x D x B – C x A x D x B + A x D x B x C  – D x A x B x C – A x D x C x B  + D x A x C x B +  C  x D x A x B –   C x D x B x A –  D x C x A x B + D x C x B x A  – B x D x A x C  + B x D x C x A   + D x B x A x C  – D x B x C x A  + B x C x A x D  –  B x C x D x A  –  C x B x A x D + C x B x D x A = 0 .

In other words, we take the sum of the products of the matrices for all 24 possible orderings (permutations) with certain plus or minus signs, and lo and behold, we always get 0.

I will say more about it. But first a few remarks. The Amitsur-Levitzki theorem deals with products of 2k matrices of size k \times k. It is very beautiful and important and when it comes to mathematics, it doesn’t get much better than that. It can be a nice theorem to explain to non mathematicians, but in this case I have especially one non-mathematician in mind. Alex Levitzki - Yaacov Levitzkii’s son.  I promised Alex who is a famous HU biologist and chemist to tell him about his father’s theorem so why not share it with others. Yaacov Levitzki was one of the founding members of my department in Jerusalem. As a young man he came to Göttingen with the idea to study chemistry but attending a lecture by Emmy Noether converted him to mathematics.

The Amitsur Levitzki Theorem: For every 2n matrices of size n by n denoted by A_1,A_2,\dots A_{2n}, we have: 

\sum sgn (\sigma) \prod_{i=1}^{2n} A_{\sigma(i)} =0,

where the sum is taken over all (2n)! orderings (permutations) \sigma of \{1,2,\dots, 2n\} and sgn(\sigma) denotes the sign of the ordering  \sigma. Continue reading