Minimal polynomial
= Minimal polynomial
{title2=$m_A(t)$}
{wiki=Minimal_polynomial_(linear_algebra)}
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> $\begin{pmatrix}2&1\\0&2\end{pmatrix}$ 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>.