Question

5. Throughout this question, fix A as an n×n matrix. If f(x) is a polynomial, then f(A) is the expression formed by replacing every x in f(x) with A and inserting the n×n identity matrix I to its constant term. For example, if f(x) = x2 −2x+5 (whose degree is 2), then f(A) = A2 −2A+5I; if f(x) = −x3 +2 (whose degree is 3), then f(A) = −A3 + 2I.

   1. (a) Using induction of the degree of the polynomial f(x), show that f(A) is an n×n matrix. Furthermore, the following useful proposition can be proved, but you do not need to prove it.

      Proposition 1. Let f(x), g(x) and h(x) be three polynomials. • If f(x) = g(x) + h(x), then f(A) = g(A) + h(A).
      • If f(x) = g(x)h(x), then f(A) = g(A)h(A).

   2. (b) For the rest of this question, assume that f(x) is a polynomial such that f(A) = 0 (the right hand

      side is the n × n matrix whose all entries are zero) and f(x) = g(x)h(x); both g(x) and h(x) are

      polynomials. (Notice that that f(A) = 0 does not necessarily mean that f(x) = 0. For example, if 

A= 0 1 ,thenitcanbedirectlyverifiedthatf(x)=x2−1,g(x)=x−1andh(x)=x+1 10

satisfy the above assumptions.) Show that for every vector v ∈ Rn and every polynomial b(x), b(A)h(A)v ∈ ker(g(A)) and b(A)g(A)v ∈ ker(h(A)).

(c) The two polynomials g(x) and h(x) are further assumed to be coprime, which means that there are

polynomials b(x) and c(x) such that b(x)g(x) + c(x)h(x) = 1. (For example x + 1 and x − 1 are

coprime, with b(x) = 1 and c(x) = − 1 .) Show that Rn = ker(g(A)) ⊕ ker(h(A)).

Answer 1

nce m PCx) mt1 m+l muvo atl dag^ues Scanned with CamScanner GS

w-e have , 9(A) (b(A)-h(A)g) Cx) 中(N) V -30 Cs Scanned with CamScanner

t.e tur ana', e R ง;ง, t'L @wre um ajain CS Scanned with CamScanner
hence proved