Let λ1, λ2, ..., λk be all the distinct eigenvalues of an n by n matrix A, with algebraic multiplicities m1, m2, ..., mk. Then by this result, the characteristic polynomial is
p(t) = det(A - λI) = (λ1 - λ)m1 (λ2 - λ)m2 ... (λk - λ)mk.
By Cayley-Hamilton theorem, we have
p(A) = (λ1I - A)m1 (λ2I - A)m2 ... (λkI - A)mk = O.
In particular, we have
In an earlier example, we considered the following diagonalizable matrices
|A = [||13||-4||], with eigenvalues 5, 15;|
|B = [||1||3||-3||], with eigenvalues 4, -2.|
For the polynomial p(t) = (5 - t)(15 - t) = 75 - 20t + t2, applying the matrix p(A) = 75I - 20A + A2 to the basis of eigenvectors of A is the same as multiplying p(λ) = p(5 or 15) = 0 to the eigenvectors. In other words, applying p(A) to the basis vectors always give us