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An application of Cayley-Hamilton's Theorem.

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 $\deg R(x) < \deg P_A(x)$.

By Cayley-Hamilton's Theorem (v.s.  2.1), we have:

$T(A)= Q(A) \underbrace{P_A(A)}_{=0} +R(A)=R(A)$.

Noah Dana-Picard