##  [Cayley–Hamilton Theorem](/cayley-hamilton-theorem-2) 

 Definition

The statement that every square matrix satisfies its own characteristic polynomial: if p(λ) = det(λI − A) is the characteristic polynomial of a square matrix A, then p(A) = 0 (the zero matrix).

 

 

 

 

 

 





## Principle

Principle

A polynomial constructed from the determinant captures algebraic relations of a matrix; substituting the matrix into that polynomial yields an annihilating polynomial, so the characteristic polynomial annihilates the matrix.

 

 

 

 

 





## Demonstration

Demonstration

For a 2×2 matrix A = [[a, b], [c, d]], the characteristic polynomial is λ^2 − (a + d)λ + (ad − bc). Replacing λ by A gives A^2 − (trace A) A + (det A) I = 0, an explicit matrix identity.

 

 

 

 

## Misapplication

Misapplication

Applying the theorem to non-square matrices or to square matrices without attention to the base ring (for example assuming it holds verbatim over rings where determinants are not well-behaved) leads to incorrect conclusions.

 

 

 

 

 





## Consequence

Consequence

Ensures the characteristic polynomial is an annihilating polynomial and implies the minimal polynomial divides the characteristic polynomial; it underpins computations of matrix functions, derivation of identities involving trace and determinant, and structural decompositions.

 

 

 

 

## Reversal

Reversal

The converse — that every annihilating polynomial must be a multiple of the characteristic polynomial — is false in that not every annihilating polynomial equals the characteristic polynomial, but the minimal polynomial always divides the characteristic polynomial.

 

 

 

 

 





## Boundary

Boundary

Holds for square matrices over commutative rings or fields where the characteristic polynomial is defined via determinant; caution is needed over noncommutative coefficient rings or in contexts lacking a determinant theory.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Competes with the viewpoint of the minimal polynomial and Jordan canonical form: Cayley–Hamilton gives a universal annihilator (the characteristic polynomial), while the minimal polynomial captures the smallest annihilating relation and more refined structure.

 

 

 

 

 





## Synthesis

Synthesis

Cayley–Hamilton unites determinant-based spectral data with algebraic relations of matrices: the characteristic polynomial, computed from trace and determinant data, when evaluated at the matrix, yields the zero operator and links eigenvalue information to polynomial functional calculus.