Cayley-Hamilton theorem

Here is one of the great surprises of linear algebra. Take an operator T, write down its characteristic polynomial chi(x) = det(xI - T), and then — instead of plugging in a number for x — plug in the operator T itself. The Cayley-Hamilton theorem says you always get zero: chi(T) = 0. Every operator satisfies its own characteristic equation.

There is a famous WRONG proof: 'just substitute T into chi(x) = det(xI - T) to get det(TI - T) = det(0) = 0.' That is nonsense, because det(xI - T) is a scalar-valued function of the number x, while chi(T) is an operator; you cannot substitute a matrix into the slot where a scalar lived. The real proof uses the adjugate identity (xI - T) adj(xI - T) = chi(x) I and matches polynomial coefficients on both sides, or invokes the dense set of diagonalizable matrices.

The single most useful corollary: the minimal polynomial DIVIDES the characteristic polynomial. Cayley-Hamilton hands you one explicit annihilating polynomial of degree n, namely chi; since the minimal polynomial divides every annihilator, it divides chi. Together with the fact that they share roots, this squeezes the minimal polynomial into a small, computable set of candidates.

Practically, Cayley-Hamilton lets you reduce any high power of T to a polynomial of degree below n: rewrite T^n in terms of lower powers using chi(T) = 0, and recurse. It is the engine behind computing T^k, the matrix exponential, and — when T is invertible — even T^-1 as a polynomial in T.