Diagonalizability and Topology In an exchange with one of my math professors, I was told that diagonalizability can differ depending on whether one is dealing with the field of real numbers or the complex numbers.
I learned in Real Analysis last term that concepts like "compactness" similarly vary over different fields.
My question given the above is simply this - is diagonalizability of matrices  (or their underlying linear systems) also a topological concept, or related to topology in some way?  I have not taken topology (only linear algebra) so I apologize in advance if this is an obvious/stupid question.
 A: Well, sort of. In order for a square matrix $M$ to be diagonalizable over a field $F$ its eigenvalues need to exist over $F$; equivalently, its characteristic polynomial needs to have all of its roots in $F$. $M$ can have any (monic) characteristic polynomial, thanks to the existence of companion matrices. Over $\mathbb{R}$ there are matrices such as rotation matrices
$$R_{\theta} = \left[ \begin{array}{cc} \cos \theta & \sin \theta \\ - \sin \theta & \cos \theta \end{array} \right]$$
which don't have real eigenvalues (for most values of $\theta$), or equivalently whose characteristic polynomials don't have roots over $\mathbb{R}$. However, by the fundamental theorem of algebra every polynomial with real or complex coefficients has all of its roots over $\mathbb{C}$, so this difficulty disappears over $\mathbb{C}$ (e.g. all rotation matrices are diagonalizable).
The connection to topology is that many proofs of the fundamental theorem of algebra make use of topological ideas, and some of them explain why FTA is true for $\mathbb{C}$ but not $\mathbb{R}$ in terms of topological differences between them, the most basic one being that $\mathbb{R}$ is $1$-dimensional but $\mathbb{C}$ is $2$-dimensional. One such proof involves applying the Lefschetz fixed point theorem to the complex projective spaces $\mathbb{CP}^n$; if you try to apply the proof to the real projective spaces $\mathbb{RP}^n$ it will succeed in odd dimensions (corresponding to the fact that every polynomial of odd degree over $\mathbb{R}$ has a real root) but fail in even dimensions (a polynomial of even degree over $\mathbb{R}$ can have no real roots).
