Take a square matrix A of order n and a polynomial T(x) of degree r, such that r>n. How can we compute T(A)?
Of course a direct computation is always possible, but perhaps not so illuminating.
Denote by PA(x) the characteristic polynomial of A and then use Euclide's algorithm: there exists a unique ordered pair of polynomials (Q(x),R(x)) such that T(x)=Q(x) PA(x) +R(x) and .
By Cayley-Hamilton's Theorem (v.s. 2.1), we have: