Hi guys.

I have the following matrix:

its characteristic polynomial is:

I've been asked if the above matrix has diagonal matrix over , I wrote "no, since the geometric multiplicity of the eigenvalue 1 is 1 (and thus smaller then the algebraic multiplicity).

and then, I've been asked if the above matrix has diagonal matrix over .

I'm guessing no, but I came here to ask, just to make sure I'm not missing anything here:

why does it make a difference if we're over or in this case?

Is it possible to get a different geometric multiplicity in different fields, for the same roots?