How to prove that there aren't square matrices $A$ and $B$, such as $AB - BA = I_n$

Question

I don't know if the matrices $AB$ and $BA$ are invertible, nor do I know if $A$ or $B$ are invertible. I also don't have the matrices to check.

The only thing I know is that they're square $_n._n$.

With this information, how can I prove that there aren't matrices $A$,$B$, such as $AB - BA = I_n$?

Fun fact: this plays a role in quantum mechanics, showing that infinite dimensional spaces are quite necessary! — Cameron L. Williams, Oct 29 '19 at 14:54

score 3 · Accepted Answer · answered Oct 29 '19 at 14:45

3

Hint (for appropriate field assumption): what happens when you take the trace of $AB - BA$?

answered Oct 29 '19 at 14:45

Mariah

It should be 0? – Duarte Arribas Oct 29 '19 at 14:46
@DuarteArribas right, and what's the trace of $Id$ when the characteristic is zero? – Mariah Oct 29 '19 at 14:46
If the characteristic is 0, that means that there aren't any pivots, so I'd assume also zero? – Duarte Arribas Oct 29 '19 at 14:47
@DuarteArribas the trace of $Id$ is $n$. – Mariah Oct 29 '19 at 14:48
@DuarteArribas Characteristic here means the characteristic of the field. A field of characteristic zero means that $1+1+\cdots+1$ can never equal $0$ (e.g. the field of real numbers) as opposed to, say, $\Bbb F_2$. – Cameron L. Williams Oct 29 '19 at 14:48
Ah, right, so it's n, as it is the sum of all diagonal elements of $I_n$ – Duarte Arribas Oct 29 '19 at 14:50
Ah, I think I understand, n = 0, so it has no solutions. – Duarte Arribas Oct 29 '19 at 14:59

1 Answers1