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For a square matrix over a field, the minimal polynomial $m_A$ is the unique monic polynomial of least degree satisfying $m_A(A)=0$. The Cayley-Hamilton theorem guarantees at least one such polynomial: the characteristic polynomial. The minimal polynomial divides every polynomial that annihilates $A$, so reducing matrix powers modulo $m_A$ can need fewer coefficients than reducing modulo the characteristic polynomial.

For example, $2I_2$ has minimal polynomial $t-2$, whereas the Jordan block $\left(\begin{array}{cc}2& 1\\ 0& 2\end{array}\right)$ has minimal polynomial $(t-2{)}^2$. Both have the same characteristic polynomial. Their different matrix powers are illustrated in repeated eigenvalues in the Cayley-Hamilton theorem.