# Heisenberg's Uncertainty Principle using Linear Algebra

I am working through linear algebra from MITs MOOC online courses. One of the question refers to the uncertainty principle. It states:

AB-BA=I can happen for infinite matrices with A

$$A=A^{ T }\\ and\\ B=-B^{ T }\\ Then\\ x^{ T }x=x^{ T }ABx-x^{ T }BAx\le 2\parallel Ax\parallel \parallel Bx\parallel$$

My question is how does
$$x^{ T }x=x^{ T }ABx-x^{ T }BAx$$?

bhobba
Mentor
Like you said:
AB-BA=I

Thanks
Bill

rpthomps
Like you said:

Thanks
Bill

Okay. I should have got that one. :) Okay, now I want to prove the premise AB-BA=I, is there an elegant way of doing that?

bhobba
Mentor
Okay. I should have got that one. :) Okay, now I want to prove the premise AB-BA=I, is there an elegant way of doing that?

That is the premise of the theorem except for a multiplicative constant - the I is replaced by iC - C a real constant.

Thanks
Bill

rpthomps
Thanks again I really appreciate your attention.