Let p() = (1)n(n an1n1 a1 a0) be a polynomial of degree n 1, and let C = an1 an2 a1 a0 1

Chapter 6, Problem 35

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Let p() = (1)n(n an1n1 a1 a0) be a polynomial of degree n 1, and let C = an1 an2 a1 a0 1 0 0 0 0 1 0 0 . . . 0 0 1 0 (a) Show that if i is a root of p() = 0, then i is an eigenvalue of C with eigenvector x = (n1 i , n2 i , ... , i, 1)T . (b) Use part (a) to show that if p() has n distinct roots then p() is the characteristic polynomial of C. The matrix C is called the companion matrix of p().