Jordan normal form for complex matrices

Question

Suppose we are given the  characteristic polynomial and minimal polynomial of a matrix, say, $(x-a)^4(x-b)^2$ and $(x-a)^2(x-b)$. Then, I can tell what the largest Jordan blocks are, and hence work out the possible forms the JNF can take. However, my question specifies that the matrix is "complex" whereas $a,b\in \mathbb R$. Is there some catch?

Answer 1

In general, a matrix $A$ with coefficients in a field $F$ has a Jordan Canonical Form (over $F$) if and only if the characteristic polynomial of $A$ splits over $F$.

We usually work in the context of the complex numbers because then we know that the characteristic polynomial will definitely split, no matter what $A$ is.

But in fact you can work over any field if you happen to know that your specific matrix $A$ has a characteristic polynomial that splits. That is the case here: you are told what the characteristic polynomial is, so you know that it splits. Therefore, you know for sure that $A$ has a Jordan Canonical From (over the reals; or in fact, over $\mathbb{Q}(a,b)$).