Solution Found!
Solution: Exercises 19–23 concern the polynomial and an n ×
Chapter , Problem 22E(choose chapter or problem)
Exercises 19–23 concern the polynomial
\(p(t)=a_{0}+a_{1} t+\cdots+a_{n-1} t^{n-1}+t^{n}\) and an \(n \times n\) matrix \(C_p\) called the companion matrix of p:
\(C_p=\left[\begin{array}{ccccc} 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & & 0 \\ \vdots & & & & \vdots \\ 0 & 0 & 0 & & 1 \\ -a_0 & -a_1 & -a_2 & \ldots & -a_{n-1} \end{array}\right]\)
Let \(p(t)=a_0+a_1 t+a_2 t^2+t^3\), and let \(\lambda\) be a zero of p.
a. Write the companion matrix for p.
b. Explain why \(\lambda^3=-a_0-a_1 \lambda-a_2 \lambda^2\), and show that \(\left(1, \lambda, \lambda^2\right)\) is an eigenvector of the companion matrix for p.
Questions & Answers
QUESTION:
Exercises 19–23 concern the polynomial
\(p(t)=a_{0}+a_{1} t+\cdots+a_{n-1} t^{n-1}+t^{n}\) and an \(n \times n\) matrix \(C_p\) called the companion matrix of p:
\(C_p=\left[\begin{array}{ccccc} 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & & 0 \\ \vdots & & & & \vdots \\ 0 & 0 & 0 & & 1 \\ -a_0 & -a_1 & -a_2 & \ldots & -a_{n-1} \end{array}\right]\)
Let \(p(t)=a_0+a_1 t+a_2 t^2+t^3\), and let \(\lambda\) be a zero of p.
a. Write the companion matrix for p.
b. Explain why \(\lambda^3=-a_0-a_1 \lambda-a_2 \lambda^2\), and show that \(\left(1, \lambda, \lambda^2\right)\) is an eigenvector of the companion matrix for p.
ANSWER:Solution 22E1. The 33 companion matrix for the polynomial is,= (