I am trying to solve a problem and I need to know whether I can deduce the following:
Suppose is normal, and there exist a polynomial such that . Then . The part I am not so sure about is that .
Can I say that and if so why?
Any direction will be appreciated
Can't think of a rank argument. How would you show this using ranks?
I was able to figure it out though, here's my proof.
Let's say , since is normal it follows is normal, therfore S can be represented by a diagonal matrix . We know , that means that and therfore where are the coordinates of . So for every we get . Now if we look at it can be representd by and therfore