In linear algebra, the Cayley–Hamilton theorem (named after the mathematicians Arthur Cayley and William Rowan Hamilton) states that every square matrix over a commutative ring (such as the real or complex numbers or the integers) satisfies its own characteristic equation.

If A is a given n × n matrix and I_n  is the n × n identity matrix, then the characteristic polynomial of A is defined as ${\displaystyle p_{A}(\lambda )=\det(\lambda I_{n}-A)}$, where det is the determinant operation and λ is a variable for a scalar element of the base ring. Since the entries of the matrix ${\displaystyle (\lambda I_{n}-A)}$ are (linear or constant) polynomials in λ, the determinant is also a degree-n monic polynomial in λ, ${\displaystyle p_{A}(\lambda )=\lambda ^{n}+c_{n-1}\lambda ^{n-1}+\cdots +c_{1}\lambda +c_{0}~.}$ One can create an analogous polynomial ${\displaystyle p_{A}(A)}$ in the matrix A instead of the scalar variable λ, defined as ${\displaystyle p_{A}(A)=A^{n}+c_{n-1}A^{n-1}+\cdots +c_{1}A+c_{0}I_{n}~.}$ The Cayley–Hamilton theorem states that this polynomial expression is equal to the zero matrix, which is to say that ${\displaystyle p_{A}(A)=\mathbf {0} }$. The theorem allows Aⁿ to be expressed as a linear combination of the lower matrix powers of A. When the ring is a field, the Cayley–Hamilton theorem is equivalent to the statement that the minimal polynomial of a square matrix divides its characteristic polynomial. The theorem was first proven in 1853 in terms of inverses of linear functions of quaternions, a non-commutative ring, by Hamilton. This corresponds to the special case of certain 4 × 4 real or 2 × 2 complex matrices. The theorem holds for general quaternionic matrices. Cayley in 1858 stated it for 3 × 3 and smaller matrices, but only published a proof for the 2 × 2 case. The general case was first proved by Ferdinand Frobenius in 1878.