I'm a TA with Advanced Algebra in school and teach the Jordan Form now. There are three questions about eigenvalues in this chapter:
Given matrix $A$, $B$ and polynome $f$, consider the eigenvalues' relation between
$A$ and $A^T$
$AB$ and $BA$
$A$ and $f(A)$
Except the last, we can solve them without the Jordan Form because of $|\lambda I-A|=|\lambda I-A^T|$ and $|\lambda I-AB|=|\lambda I-BA|$. Last is easily solved by Jordan Form, and the conclusion is
If $\lambda_1$, $\cdots$, $\lambda_n$ are $A$'s eigenvalues, then $f(\lambda_1)$, $\cdots$, $f(\lambda_n)$ are $f(A)$'s eigenvalues.
However I am not satisfied with the method of Jordan Form since
The first and second are just use the properties of determinant
If using the method of Jordan Form, we actually introduce the field of coefficient into its algebraically closed field. But I want to avoid the situation.
So are there any other approaches to the last question? Any advice is helpful. Thank you.