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Cayley-Hamilton says that a square matrix $A$ over any commutative ring $R$ satisfies its own characteristic equation.

Conversely, suppose that $p(T) \in R[T]$ is satisfied by $A$, i.e., that $p(A) = 0$. What is the most we can say about $p(A)$?

I know that if $m_A$ denotes the minimal polynomial of $A$, then $m_A \mid p$, so in particular, the roots of $p$ include the eigenvalues of $A$. Is there anything else one can say?

Rodrigo de Azevedo
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Drew Brady
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  • I am not sure, but I think we cannot say more in general. – Peter Feb 04 '18 at 19:38
  • Maybe this is obvious to you, but $m_A$ divides $p$ if and only if $p(A) = 0$. If we could say more, then it would apply to any polynomial that $m_A$ divides. –  Apr 23 '19 at 23:47

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