Cayley–Hamilton theorem
Introduction
The Cayley–Hamilton theorem is a fundamental theorem in linear algebra, named after the mathematicians Arthur Cayley and William Rowan Hamilton. The theorem states that every square matrix over a commutative ring (such as the real or complex numbers) satisfies its own characteristic equation.
Statement of the Theorem
The Cayley–Hamilton theorem can be stated as follows: Let A be a square n x n matrix over a field F. The characteristic polynomial p(λ) of A is defined as the determinant of the matrix (λI - A), where I is the identity matrix of the same size as A, and λ is a scalar. The Cayley–Hamilton theorem then states that p(A) = 0, where 0 is the zero matrix of the same size as A.
Proof of the Theorem
The proof of the Cayley–Hamilton theorem is typically done by induction on the size of the matrix. The base case is trivial, as a 1 x 1 matrix trivially satisfies its characteristic equation. The induction step involves a clever use of the Laplace expansion and the properties of the determinant.
Applications of the Theorem
The Cayley–Hamilton theorem has numerous applications in linear algebra and its related fields, such as differential equations and control theory. For example, it can be used to find the inverse of a matrix, to compute the powers of a matrix, and to solve systems of linear differential equations.
Generalizations and Related Results
The Cayley–Hamilton theorem can be generalized to matrices over a commutative ring, and to linear operators on a finite-dimensional vector space. There are also related results, such as the Hamilton–Cayley theorem, which states that a linear operator on a finite-dimensional vector space satisfies its minimal polynomial.