Consider a vector space $V$ over finite field $F_p$. Suppose $A$ be a square matrix. Also let $C(x)$ is a characteristic polynomial of $A$. Suppose C(x)=$\prod g_i(x)$ for $i=1:k$. It is noted that $g_i(x)$ is irreducible for all $i$.
It is true that minimal polynomial divides characteristic polynomial. Is it true that the annihilating polynomial divides minimal polynomial?
Question is: If one of the $g_i(x)$ divides the minimal polynomial, does that make the guarantee that the $g_i(x)$ is annihilating polynomial? We know the converse is true.
Best Answer
In one point you got the direction mixed up: The minimal poylnomial divides each annihilating polynomial, not the other way around.
The irreducuble factors $g_i(x)$ of $C(x)$ are in general not annihilating - this is only the case if $C(x)$ is a power of $g_i(x)$ (up to constant factor). Each irreducible factor occuring in $C(x)$ willalso occur in the minimal poylnomial, though often to a lower power.