Chapter 5, Problem 23SE

Exercises 19–23 concern the polynomial

p(t) = a₀ + a₁t + … + a_n−1tⁿ⁻¹ + tⁿ

and an n × n matrix C_p called the companion matrix of p:

C_p = $\left[\begin{array}{ccccc}\text{0}& \text{1}& \text{0}& \cdots & \text{0}\\ \text{0}& \text{0}& \text{1}& & \text{0}\\ ⋮& & & & ⋮\\ 0& \text{0}& \text{0}& & \text{1}\\ -{\text{a}}_0& -{\text{a}}_1& -{\text{a}}_2& & -{\text{a}}_{\text{n}-1}\end{array}\right]$

23. Let p be the polynomial in Exercise 22, and suppose the equation p(t) = 0 has distinct roots λ₁, λ₂, λ₃. Let V be the Vandermonde matrix

V = $\left[\begin{array}{ccc}1& 1& 1\\ {\lambda }_1& {\lambda }_2& {\lambda }_3\\ {\lambda }_1^2& {\lambda }_2^2& {\lambda }_3^2\end{array}\right]$

(The transpose of V was considered in Supplementary Exercise 11 in Chapter 2.) Use Exercise 22 and a theorem from this chapter to deduce that V is invertible (but do not compute V⁻¹). Then explain why V⁻¹C_pV is a diagonal matrix.