The Cayley–Hamilton theorem is a fundamental result in linear algebra that states that every square matrix satisfies its own characteristic polynomial.
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Every square matrix satisfies its own characteristic polynomial. More explicitly, for an -by- matrix over a commutative ring with identity, formThenHere 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.[ref]
Universal Identities by Keith Conrad. §3, Theorem 3.4 and the universal-identity discussion, printed pp. 6–7: matrices over arbitrary commutative rings
The practical payoff is that arbitrarily high matrix powers can be reduced to a combination of . 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.