Let $K$ be a field. For a matrix $A\in GL_n(K)$ we can find the Jordan normal form $A'$ in $GL_n(\overline{K})$, where $\overline{K}$ is the algebraic closure of $K$. We write $j_\alpha(A)$ for the number of $1\times 1$ Jordanblocks with eigenvalues $\alpha$ in $A'$. Now consider $\alpha\notin K$ and the corresponding number $j_\alpha(A)$. If $K$ is a perfect field, then $\overline{K}:K$ is a separable extension and for each occurence of $\alpha$ there must be at least one eigenvalue $\beta$ of $A$ which is a Galois conjugate of $\alpha$, and hence in particular $\alpha\neq \beta$. So we can conclude that $j_\alpha(A)\leq\frac{1}{2}n$. The typical example is $$\left(\begin{matrix} 0 & 1\\\ -1 &0\end{matrix}\right)$$ over $\mathbb{R}$ with complex eigenvalues $\alpha=i$, $\beta=-i$.

The question is:
**If $K$ is an imperfect field, is it still true that $j_\alpha(A)\leq \frac{1}{2}n$ if $\alpha\notin K$? If not, how can one find a counterexample?**

I tried more or less randomly computing Jordan normal forms of matrices over $\mathbb{F}_2(x^2)$ and similar fields, starting from the (non-counter)example $$\left(\begin{matrix} 0 & 1\\\ x^2 &0\end{matrix}\right)$$ with eigenvalues $x$ and $x$ (and one $2\times 2$ Jordan block). Unfortunately I am lacking ideas how to constructively build a working counterexample.