Minimal polynomial

The minimal polynomial $m_A$ is the unique monic polynomial of least degree satisfying $m_A(A)=0$. It divides every polynomial that annihilates $A$.

A square matrix $A$ is invertible exactly when its minimal polynomial has nonzero constant term. If $m_A(t)=tq(t)$, then $Aq(A)=0$, and invertibility would contradict the minimality of $m_A$. Conversely, if $m_A(t)=tq(t)+c$ with $c\ne 0$, then $A[-q(A)/c]=I$.

For square matrices $A,B$,Indeed, $(BA{)}^{k+1}=B(AB{)}^{k}A$. Thus the two minimal polynomials have the same nonzero factors with the same multiplicities, while their powers of $t$ differ by at most one.

If $p_1,\dots ,p_r$ are pairwise coprime, thenBezout identities between the factors construct the corresponding projections.

If an upper-triangular $n$ by $n$ matrix has diagonal entries ${\lambda }_1,\dots ,{\lambda }_n$, its standard invariant flag satisfiesThe commuting product ${\prod }_k(A-{\lambda }_{k}I)$ therefore vanishes, proving $\mathrm{deg}{m}_A\le n$ without invoking the full Cayley-Hamilton theorem.