Exercises 19–23 concern the polynomial and an n × n
Chapter , Problem 23E(choose chapter or problem)
Exercises 19–23 concern the polynomial and an n × n matrix Cp called the companion matrix of p: Let p be the polynomial in Exercise 22, and suppose the equation p(t) = 0 has distinct roots . Let V be the Vandermonde matrix (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–1 CpV is a diagonal matrix.Reference Exercise 22:Exercises 19–23 concern the polynomial and an n × n matrix Cp called the companion matrix of p: a. Write the companion matrix for p. and show that is an eigenvector of the companion matrix for p.
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