Old qual question here:

Suppse $(V,\langle,\rangle)$ is a Hermitian inner product space, and $T:V\to V$ is a complex linear transformation. Show that the following are equivalent:

  1. For all $v\in V$, $\langle Tv,Tv\rangle=\langle v,v\rangle$
  2. For all $v,w\in V$, $\langle Tv,Tw\rangle=\langle v,w\rangle$

Now obviously $2\implies1$ but I'm not sure how to go the other way.


Hint. Remember the polarization identity: $$\langle v,w\rangle=\frac{1}{4}(\langle v+w,v+w\rangle-\langle v-w,v-w\rangle+i\langle v+iw,v+iw\rangle-i\langle v-iw,v-iw\rangle)$$

  • $\begingroup$ So then it's as simple as replacing $v$ with $Tv$ and $w$ with $Tw$, using that $T$ is linear (so $Tv+Tw=T(v+w)$, et cetera) and then we have $\langle T(v+w),T(v+w)\rangle=\langle v+w,v+w\rangle$ (and similarly for the other three) so you get what you need? $\endgroup$ Dec 9 '16 at 17:35
  • $\begingroup$ Yes, that's exactly it! $\endgroup$
    – C. Falcon
    Dec 9 '16 at 17:40

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.