Cayley-Hamilton Theorem — Definition, Formula & Examples

The Cayley-Hamilton Theorem states that every square matrix satisfies its own characteristic polynomial. In other words, if you compute the characteristic polynomial of a matrix and then substitute the matrix itself into that polynomial, the result is the zero matrix.

Let $A$ be an $n \times n$ matrix over a field $F$ , and let $p(\lambda) = \det(\lambda I - A)$ be its characteristic polynomial. Then $p(A) = O$ , where $O$ is the $n \times n$ zero matrix. That is, $\det(\lambda I - A)\big|_{\lambda = A} = O$ , with the constant term $c_0$ interpreted as $c_0 I$ .

Key Formula

p(A) = A^n + c_{n-1}A^{n-1} + \cdots + c_1 A + c_0 I = O

Where:

$A$ = An n × n square matrix
$p(\lambda)$ = The characteristic polynomial det(λI − A)
$c_0, \ldots, c_{n-1}$ = Coefficients of the characteristic polynomial
$I$ = The n × n identity matrix
$O$ = The n × n zero matrix

How It Works

To apply the theorem, first find the characteristic polynomial

p(\lambda) = \det(\lambda I - A)

. Then replace every occurrence of

\lambda^k

with

A^k

and every constant

c

with

cI

. The theorem guarantees the resulting matrix expression equals the zero matrix. This identity is often used to express

A^{-1}

in terms of lower powers of

A

, or to reduce high powers of

A

to combinations of

I, A, \ldots, A^{n-1}

Worked Example

Problem: Verify the Cayley-Hamilton Theorem for A = [[1, 2], [3, 4]].

Find the characteristic polynomial: Compute det(λI − A).

p(\lambda) = \det\begin{pmatrix} \lambda - 1 & -2 \\ -3 & \lambda - 4 \end{pmatrix} = (\lambda - 1)(\lambda - 4) - 6 = \lambda^2 - 5\lambda - 2

Substitute the matrix into p: Replace λ² with A², λ with A, and the constant with −2I.

p(A) = A^2 - 5A - 2I

Compute A²: Multiply A by itself.

A^2 = \begin{pmatrix}1&2\\3&4\end{pmatrix}\begin{pmatrix}1&2\\3&4\end{pmatrix} = \begin{pmatrix}7&10\\15&22\end{pmatrix}

Evaluate p(A): Combine the matrices.

p(A) = \begin{pmatrix}7&10\\15&22\end{pmatrix} - \begin{pmatrix}5&10\\15&20\end{pmatrix} - \begin{pmatrix}2&0\\0&2\end{pmatrix} = \begin{pmatrix}0&0\\0&0\end{pmatrix}

Answer: p(A) equals the zero matrix, confirming the Cayley-Hamilton Theorem.

Why It Matters

The Cayley-Hamilton Theorem lets you express the inverse of a matrix as a polynomial in that matrix, which is the basis for computational methods in control theory and differential equations. It also shows that every matrix power

A^n

and beyond can be written as a linear combination of

I, A, \ldots, A^{n-1}

, simplifying calculations involving matrix exponentials.

Common Mistakes

Mistake: Substituting the matrix entries directly into det(λI − A) instead of replacing λ with the matrix A.

Correction: The theorem requires you to substitute the entire matrix A for the scalar λ, replacing each λ^k with the matrix power A^k and each constant c with cI.