= Cayley-Hamilton theorem
{c}
{title2=$\chi_A(A)=0$}
{wiki}
Every <square matrix> satisfies its own <characteristic polynomial>. More explicitly, for an $n$-by-$n$ <matrix> $A$ over a <commutative ring> with identity, form
$$
\chi_A(t)=\det(tI-A)=t^n+c_{n-1}t^{n-1}+\cdots+c_1t+c_0.
$$
Then
$$
\boxed{A^n+c_{n-1}A^{n-1}+\cdots+c_1A+c_0I=0.}
$$
Here $0$ is the <zero matrix>, and the constant term multiplies the <identity matrix>. This is a <polynomial evaluated at a matrix>; it is stronger than saying that each <eigenvalue> is a <root of a polynomial>. Neither <diagonalizability> nor distinct <eigenvalues> is required.<Universal Identities by Keith Conrad>[§3, Theorem 3.4 and the universal-identity discussion, printed pp. 6–7: matrices over arbitrary commutative rings]{ref}
The practical payoff is that arbitrarily high <matrix powers> can be reduced to a combination of $I,A,\ldots,A^{n-1}$. These powers arise when repeatedly applying a <linear map>, solving a <linear recurrence relation>, or evolving a finite <state space> model. The <characteristic polynomial> turns that repeated action into a fixed-order scalar recurrence. The <minimal polynomial> may reduce the number of required powers further.
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