Cayley–Hamilton Theorem

Statement

Let $A$ be an $n \times n$ matrix over a commutative ring $R$, and let $p_A(\lambda) = \det(\lambda I - A)$ be its characteristic polynomial. Then:

That is, substituting $A$ itself for the scalar variable $\lambda$ in the characteristic polynomial yields the zero matrix.

Visualization

Take the concrete $2 \times 2$ matrix:

Step 1 — Compute the characteristic polynomial.

Step 2 — Compute $A^2$.

Step 3 — Evaluate $p_A(A) = A^2 - 7A + 10I$.

Verified: $p_A(A) = 0$.

Proof Sketch

The naive approach — "substitute $A$ for $\lambda$ in $\det(\lambda I - A)$" — is circular: the determinant expands as a scalar polynomial, not a matrix polynomial. The correct proof works over the adjugate matrix $\text{adj}(\lambda I - A)$, whose entries are polynomials in $\lambda$. One shows $(\lambda I - A)\,\text{adj}(\lambda I - A) = p_A(\lambda)\,I$ as a polynomial matrix identity, then compares coefficients of $\lambda^k$ on both sides to assemble the relation $p_A(A) = 0$.

An alternative proof observes that the result holds over any field (by finding a splitting field where $A$ is upper-triangular) and then descends to general commutative rings by a universality argument.

Connections

• Spectral TheoremSpectral Theorem$$A = Q \Lambda Q^T \;\text{(real symmetric case)}$$Every real symmetric matrix is orthogonally diagonalizableRead more → — eigenvalues are roots of $p_A$; Cayley–Hamilton shows $A$ itself is a "root."
• Determinant MultiplicativityDeterminant Multiplicativity$$\det(AB) = \det(A)\,\det(B)$$The determinant of a product equals the product of determinantsRead more → — the characteristic polynomial is built from the determinant.
• Rank–Nullity TheoremRank–Nullity Theorem$$\operatorname{rank}(f) + \operatorname{nullity}(f) = \dim V$$The dimensions of image and kernel of a linear map sum to the domain dimensionRead more → — the minimal polynomial divides $p_A$; its degree relates to the kernel.
• Fundamental Theorem of Galois TheoryFundamental Theorem of Galois Theory$$\{\text{intermediate fields}\} \longleftrightarrow \{\text{subgroups of } \text{Gal}(K/F)\}$$Bijection between intermediate fields and subgroups of the Galois groupRead more → — Galois extensions are controlled by characteristic polynomials of Frobenius elements.
• Quadratic FormulaQuadratic Formula$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$Solve any quadratic equation using the discriminantRead more → — in dimension 2, $p_A$ is quadratic; Cayley–Hamilton is the matrix analogue of a root equation.

/-- The Cayley–Hamilton theorem: every square matrix satisfies its own
    characteristic polynomial. The Mathlib proof works over any commutative
    ring via the adjugate identity. -/
theorem cayley_hamilton {n : Type*} {R : Type*}
    [Fintype n] [DecidableEq n] [CommRing R]
    (A : Matrix n n R) : Polynomial.aeval A A.charpoly = 0 :=
  Matrix.aeval_self_charpoly A