##  [Cayley–Hamilton Theorem](/cayley-hamilton-theorem-1) 

 Definition

The statement that every square matrix over a commutative ring (in particular over a field) satisfies its own characteristic polynomial: if p(λ) = det(λI − A) then p(A)=0 when p is evaluated with matrix substitution.

 

 

 

 

 

 





## Principle

Principle

Substituting the matrix into its characteristic polynomial yields the zero matrix because the polynomial annihilates the linear transformation represented by the matrix.

 

 

 

 

 





## Demonstration

Demonstration

For a 2×2 matrix A = [[a,b],[c,d]], its characteristic polynomial p(λ)=λ^2−(a+d)λ+(ad−bc); Cayley–Hamilton asserts p(A)=A^2−(a+d)A+(ad−bc)I = 0, an explicit matrix identity that can be checked by calculation.

 

 

 

 

## Misapplication

Misapplication

Plugging A into a polynomial with coefficients that do not commute with A (noncommutative coefficient ring) without checking centrality, or confusing evaluation of scalar polynomials at A with pointwise substitution of scalars.

 

 

 

 

 





## Consequence

Consequence

Allows expressing higher powers of A as lower-degree polynomials in A, provides a route to compute matrix functions and inverses (when invertible) by polynomial reduction, and links characteristic and minimal polynomials.

 

 

 

 

## Reversal

Reversal

The converse is not automatic: a polynomial q with q(A)=0 need not equal the characteristic polynomial; it must divide the minimal polynomial, which in turn divides the characteristic polynomial.

 

 

 

 

 





## Boundary

Boundary

Standard Cayley–Hamilton holds for matrices over commutative rings and fields; extensions to matrices over noncommutative rings require care about the order of multiplication and coefficient centrality. It presupposes a well-defined characteristic polynomial.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension exists between the characteristic polynomial and the minimal polynomial: Cayley–Hamilton guarantees annihilation by the characteristic polynomial, but the minimal polynomial is the smallest annihilating polynomial and is often the sharper invariant for dynamics.

 

 

 

 

 





## Synthesis

Synthesis

Cayley–Hamilton unifies linear algebraic structure by asserting that the characteristic polynomial annihilates its matrix, enabling algebraic reduction of matrix powers and providing a bridge between determinant-based invariants and operator-theoretic annihilators.