(a) Use mathematical induction to prove that, for n 2, the companion matrix C( p) of

Chapter 4, Problem 4.3.32

(choose chapter or problem)

(a) Use mathematical induction to prove that, for n 2, the companion matrix C( p) of p(x) x n an1x n1 % a1 x a0 has characteristic polynomial (1) n p(l). [Hint: Expand by cofactors along the last column. You may find it helpful to introduce the polynomial q(x) ( p(x) a0)x.] (b) Show that if l is an eigenvalue of the companion matrix C( p) in equation (4), then an eigenvector corresponding to l is given by If p(x) x n an1x n1 p a1x a0 and A is a square matrix, we can define a square matrix p(A) by An important theorem in advanced linear algebra says that if cA(l) is the characteristic polynomial of the matrix A, then cA(A) O (in words, every matrix satisfies its characteristic equation). This is the celebrated Cayley-Hamilton Theorem, named after Arthur Cayley (18211895), pictured below, and Sir William Rowan Hamilton (see page 2). Cayley proved this theorem in 1858. Hamilton discovered it, independently, in his work on quaternions, a generalization of the complex numbers. p1A2