Theorem Statement The Cayley–Hamilton Theorem says: p (A) = An + an−1 An−1 + ⋯ + a1 A + a0 In = 0 That is, if you substitute the matrix A into its own characteristic polynomial, the result is always the zero matrix. The roots of this polynomial are the eigenvalues of the matrix. Learn the proof and applications of the Cayley-Hamilton theorem, which states that the characteristic polynomial of a matrix vanishes at that matrix. See how to decompose the kernel of the characteristic polynomial into generalized eigenspaces using polynomials that are relatively prime. Cayley-Hamilton theorem by Marco Taboga, PhD The Cayley-Hamilton theorem shows that the characteristic polynomial of a square matrix is identically equal to zero when it is transformed into a polynomial in the matrix itself. In other words, a square matrix satisfies its own characteristic equation. Learn the Cayley Hamilton Theorem with a clear statement, step-by-step proof, essential formulas, and solved examples. Understand how matrices satisfy their own characteristic equations.
