Cayley–Hamilton theorem

Every square matrix satisfies its characteristic polynomial when the polynomial’s variable is replaced by the matrix. It applies to matrices over commutative rings as well as fields.

Characteristic polynomial: The polynomial det(tI−A) associated with a square matrix A. The theorem’s central ingredient is the polynomial that the matrix must satisfy.
Square matrix: A matrix with the same number of rows and columns, representing an endomorphism after choosing a basis. The theorem applies to square matrices because their characteristic polynomials are defined.
Matrix power: A product of repeated copies of a square matrix, such as A² or Aⁿ. The characteristic-polynomial relation reduces sufficiently high powers to combinations of lower powers.
Eigenvalue test for matrix polynomials: The implication that p(A)=0 forces p(λ)=0 for every eigenvalue λ of A. This consequence checks eigenvalues but does not by itself establish the full matrix identity.
Arthur Cayley: A British mathematician who helped establish matrix algebra and made major contributions to geometry and invariant theory. Cayley published an early statement and verification of the theorem for matrices.
Matrix polynomial: A polynomial whose coefficients are matrices, or a scalar polynomial evaluated at a matrix. Substituting a matrix for the variable defines the theorem’s conclusion.
Eigenvalue: A scalar λ for which a linear transformation has a nonzero vector v satisfying Av=λv. Eigenvalues are roots of the characteristic polynomial, though the theorem is stronger than its eigenvalue consequences.
Matrix exponential: The matrix-valued function exp(A)=∑ₙ₌₀∞ Aⁿ/n!, defined by a convergent power series. Power reduction can simplify computations of the exponential for a fixed finite-dimensional matrix.
Diagonalization: The representation of a matrix as similar to a diagonal matrix. The theorem holds even for matrices that cannot be diagonalized.
William Rowan Hamilton: An Irish mathematician known for the quaternions and foundational work in mechanics and algebra. Hamilton’s work on algebraic operators is part of the theorem’s historical naming and development.

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