For Linear operators $V$ and $W$, $VW-WV\neq I$

Question

Let $V$ and $W$ be linear operators on a finite dimensional vector space over a field of chararacteristic $0$. Then $VW-WV\neq I$.

Generalization: http://math.stackexchange.com/questions/54397/the-identity-cannot-be-a-commutator-in-a-banach-algebra. — Martín-Blas Pérez Pinilla, May 08 '14 at 07:06

score 3 · Answer 1 · answered Apr 22 '13 at 06:39

3

Note that $\mathrm{tr}(VW-WV)=\mathrm{tr}(VW)-\mathrm{tr}(WV)=\mathrm{tr}(VW)-\mathrm{tr}(VW)=0$ so $VW-WV\neq I$.

answered Apr 22 '13 at 06:39

Brian Fitzpatrick

1 Answers1