Relationships between matrices based on their minimal polynomials Given two matrices $A$ and $B$ with minimal polynomials $\mu _A(x)$, $\mu _B(x)$ such that $\mu _b (x)= \mu _A (x)p(x)$, does this imply any relationship between $A$ and $B$ other than that all of the eigenvalues of $A$ are also eigenvalues of $B$ (with possibly different multiplicities)? Could we construct $B$ from $A$ and $p(x)$?
I know this may be a little bit open ended, but I'd appreciate any thoughts you have on this.
Thanks! 
 A: As the dimension goes up, you know less and less about the relationship. The minimal polynomial tells you the size of the largest Jordan block with that eigenvalue. It does not tell you how many blocks are of that size. The geometric multiplicity $G$ is the number of blocks, but again not how how many of each size. So you are considering the partitions of the algebraic multiplicity $A$ into $G$ natural numbers with maximum size $S.$ 
Give me a minute to think of an example of one eigenvalue with several choices, meanwhile:
Alright, $A=15, G=6, S = 5,$ we get
$$  15 = 5 + 5 + 2 + 1 + 1 + 1 = 5 + 4 + 3 + 1 + 1 + 1,    $$
$$ 15 =  5 + 3 + 3 + 2 + 1 + 1 = 5 + 3 + 2 + 2 + 2 + 1,$$
$$ 15 = 5 + 2 + 2 + 2 + 2 + 2.  $$
Those may be all. You've got to have at least one 5, you cannot have two 4's or three 3's, and if you have no 4's or 3's you must fill in with all 2's. Actually it would not be hard to program a computer and find all these partitions, see if any are missed.
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