Exercises 19–23 concern the polynomial
p(t) = a0 + a1t + … + an−1tn−1 + tn
and an n × n matrix Cp called the companion matrix of p:
Cp =
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 =
(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−1CpV is a diagonal matrix.
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