Minimal polynomial of direct sum

Question

I know that the characteristic polynomial of the direct sum of matrices is the product of their respective characteristic polynomials.

Is it also true for the minimal polynomial?

Is there a possible way to get $\mu_{A\oplus B}$ from knowing $\mu_A$ and $\mu_B$?

Answer 1

If you apply a polynomial to $A \oplus B$ you get $f(A \oplus B) = f(A) \oplus f(B)$.

If $f(A \oplus B) = 0$ then $f(A) = f(B) = 0$ so $\mu_A \mid f$ and $\mu_B \mid f$. Conversely, if $\mu_A \mid f$ and $\mu_B \mid f$ then $f(A \oplus B) = 0$. This tells us that

$$ \mu_{A \oplus B} = \operatorname{lcm}(\mu_A, \mu_B). $$

Answer 2

No. Take $A=B=(1)$. The minimal polynomial of $A$ and $B$ is $x-1$. But the minimal polynomial of $A\oplus B$ is also $x-1$.