Cayley-Hamilton theorem

The Cayley-Hamilton theorem states that every square matrix satisfies its own characteristic equation. In simpler terms, if you compute a polynomial based on the matrix's properties, and then substitute the matrix itself into that polynomial, the result is the zero matrix. This relationship reveals a fundamental link between a matrix and its characteristic polynomial, providing a way to express the matrix as a combination of its powers, which simplifies many calculations in linear algebra and helps understand the matrix’s behavior.