Cayley–Hamilton theorem

Arthur Cayley

William Rowan Hamilton portrait oval combined

GeorgFrobenius

Cayley–Hamilton theorem is a fundamental statement in linear algebra which asserts that every square matrix over a commutative ring satisfies its own characteristic equation. Named after the mathematicians Arthur Cayley and William Rowan Hamilton, the theorem has significant implications in various areas of mathematics, including matrix theory, eigenvalues and eigenvectors analysis, and the development of polynomial functions related to matrices.

Statement of the Theorem

The Cayley–Hamilton theorem can be formally stated as follows: Given an n×n square matrix A over a commutative ring, if p(λ) = det(A - λI) is the characteristic polynomial of A, where λ is a scalar, I is the n×n identity matrix, and det denotes the determinant, then p(A) = 0. Here, 0 denotes the zero matrix of the same size as A, and p(A) is obtained by substituting the matrix A into the polynomial p.

Proof and Implications

The proof of the Cayley–Hamilton theorem involves the use of the adjugate matrix and properties of the determinant. It shows that by substituting the matrix A into its characteristic polynomial, one indeed obtains the zero matrix. This result is surprising to many, as it implies that matrices can, in a sense, be solutions to their own characteristic equations.

The implications of the Cayley–Hamilton theorem are vast. It provides a method to compute the powers of matrices and the inverse of matrices (when they exist) using the coefficients of the characteristic polynomial. This is particularly useful in solving systems of linear equations, analyzing linear dynamical systems, and in the field of control theory.

Applications

One of the key applications of the Cayley–Hamilton theorem is in the computation of matrix functions, such as the matrix exponential. This is crucial in solving systems of linear differential equations, which appear frequently in physics and engineering. Additionally, the theorem aids in simplifying expressions involving matrices, thereby making certain calculations more tractable.

See Also

• Matrix
• Determinant
• Polynomial
• Eigenvalues and eigenvectors

Further Reading

While this article provides an overview of the Cayley–Hamilton theorem, readers interested in a deeper understanding may explore topics in linear algebra that discuss the theorem's proof, its implications, and applications in more detail.