The Cayley-Hamilton theorem

The Cayley-Hamilton theorem states that any square matrix (a grid of numbers) can be expressed using its own characteristic polynomial, which is a mathematical expression derived from the matrix itself. Essentially, the theorem asserts that if you take a square matrix and substitute it into its own characteristic polynomial, the result will equal the zero matrix. This means that the matrix satisfies its own polynomial, linking algebra and linear transformations in a powerful way. It helps in understanding the behavior of systems represented by matrices, especially in fields like engineering and physics.