What does mean that two matrices are similar in terms of its minimal polynomial?

Question

Let $p_T(x) = (x-3)^2(x-7)^4(x-1)^3$ and $p_H(x) = (x-3)^4(x-7)^3(x-1)^2$ two minimal polynomials of the operators $T$ and $H$?

What does mean to define if $T$ and $H$ are similar just looking at these two polynomials?

Answer 1

Similar matrices have the same minimal polynomial and the same characteristic polynomial, but the converse is not true: two non-similar matrix can share the same minimal and characteristic polynomials.

Actually, similar matrices are characterised by the so-called similarity invariants, based on Frobenius decomposition. Any endomorphism $f$ of a finite dimensional vector space $E$ over a field $K$ makes this vector space a module over the P.I.D. $K[X]$, scalar multiplication being defined by $$X\cdot v=f(v).$$ It turns out this module is a finitely generated torsion module. As all such modules, it has invariant factors $\;p_1,\dots,p_r$ such that for each $i=1, \dots, r-1$, $p_i\mid p_{i+1}$.

In the case of the module associated to the pair $(E,u)$, these invariant factors are called the similarity invariants of the endomorphism. The last invariant is the minimal polynomial of $u$, and the product $\;p_1\dotsm p_r\;$ is its characteristic polynomial.