In Shilov's Linear Algebra, he defines an elementary divisor of the operator $A$ as follows,
$$E_p(\lambda)=\frac{D_{p+1}(\lambda)}{D_{p}(\lambda)}$$
where $D_{p}(\lambda)$ is the greatest common divisor of polynomials generated by minors of order $p$ of the matrix $A - \lambda I$.
How would we use this to actually calculate the elementary divisors from this? Do we actually have to calculate all the minors of order $p$ and then find the GCD?
Most other texts talk about generalized eigenvectors and generalized eigenspaces. How is that related to the above?