Question 7.E.3.7: Suppose that for nondiagonalizable matrices Am×m an infinite......
Suppose that for nondiagonalizable matrices A_{m×m} an infinite series f(z) = \sum_{n=0}^{∞} c_n(z − z_0)^n is used to define f(A) = \sum_{n=0}^{∞} c_n(A − z_0I)^n as suggested in (7.3.7).
f(A) = \sum\limits_{n=0}^{∞} c_n(A − z_0I)^n. (7.3.7)
Neglecting convergence issues, explain why there is a polynomial p(z) of at most degree m − 1 such that f(A) = p(A).
The Cayley–Hamilton theorem says that each A_{m×m} satisfies its own characteristic equation, 0 = det (A − λI) = λ^m + c_1 λ^{m−1} + c_2 λ^{m−2} + · · · + c_{m−1} λ + c_m, so A^m = −c_1 A^{m−1} − · · · − c_{m−1} A − c_m I. Consequently, A^m and every higher power of A is a polynomial in A of degree at most m−1, and thus any expression involving powers of A can always be reduced to an expression involving at most I, A, . . . , A^{m−1}.