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Chapter 5: Q20SE (page 267)

Exercises 19鈥�23 concern the polynomial \(p\left( t \right) = {a_{\bf{0}}} + {a_{\bf{1}}}t + ... + {a_{n - {\bf{1}}}}{t^{n - {\bf{1}}}} + {t^n}\) and \(n \times n\) matrix \({C_p}\) called the companion matrix of \(p\): \({C_p} = \left( {\begin{aligned}{*{20}{c}}{\bf{0}}&{\bf{1}}&{\bf{0}}&{...}&{\bf{0}}\\{\bf{0}}&{\bf{0}}&{\bf{1}}&{}&{\bf{0}}\\:&{}&{}&{}&:\\{\bf{0}}&{\bf{0}}&{\bf{0}}&{}&{\bf{1}}\\{ - {a_{\bf{0}}}}&{ - {a_{\bf{1}}}}&{ - {a_{\bf{2}}}}&{...}&{ - {a_{n - {\bf{1}}}}}\end{aligned}} \right)\).

20. Let \(p\left( t \right){\bf{ = }}\left( {t{\bf{ - 2}}} \right)\left( {t{\bf{ - 3}}} \right)\left( {t{\bf{ - 4}}} \right){\bf{ = - 24 + 26}}t{\bf{ - 9}}{t^{\bf{2}}}{\bf{ + }}{t^{\bf{3}}}\). Write the companion matrix for \(p\left( t \right)\), and use techniques from chapter \({\bf{3}}\) to find the characteristic polynomial.

Short Answer

Expert verified

The characteristic polynomial of the matrix \({C_p}\)is\( - p\left( \lambda \right)\).

Step by step solution

01

Step 1: Find the companion matrix

Considerthe polynomial \(p\left( t \right) = {a_0} + {a_1}t + ... + {a_{n - 1}}{t^{n - 1}} + {t^n}\).

The companion matrix of \(p\)is\({C_p} = \left( {\begin{aligned}{*{20}{c}}0&1&0&{...}&0\\0&0&1&{}&0\\:&{}&{}&{}&:\\0&0&0&{}&1\\{ - {a_0}}&{ - {a_1}}&{ - {a_2}}&{...}&{ - {a_{n - 1}}}\end{aligned}} \right)\).

Find the companion matrix for the given polynomial.

\(p\left( t \right) = - 24 + 26t - 9{t^2} + {t^3}\)

Therefore, on comparison we get,

\({C_p} = \left( {\begin{aligned}{*{20}{c}}0&1&0\\0&0&1\\{24}&{ - 26}&9\end{aligned}} \right)\)

02

Find the characteristic polynomial

\(\begin{aligned}{c}\det \left( {{C_p} - \lambda I} \right) &= \det \left( {\begin{aligned}{*{20}{c}}{ - \lambda }&1&0\\0&{ - \lambda }&1\\{24}&{ - 26}&{9 - \lambda }\end{aligned}} \right)\\ &= \left( { - \lambda } \right)\left( {\left( { - \lambda } \right)\left( {9 - \lambda } \right) + 26} \right) + 24\\ &= \left( { - \lambda } \right)\left( { - 9\lambda + {\lambda ^2} + 26} \right) + 24\\ &= 9{\lambda ^2} - {\lambda ^3} - 26\lambda + 24\\ &= - \left( { - 24 + 26\lambda - 9{\lambda ^2} + {\lambda ^3}} \right)\\ &= - p\left( \lambda \right)\end{aligned}\)

Therefore, the characteristic polynomial of the matrix \({C_p}\)is\( - p\left( \lambda \right)\).

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Most popular questions from this chapter

Question: Find the characteristic polynomial and the eigenvalues of the matrices in Exercises 1-8.

  1. \(\left[ {\begin{array}{*{20}{c}}2&7\\7&2\end{array}} \right]\)

Let\(B = \left\{ {{{\bf{b}}_{\bf{1}}},{{\bf{b}}_{\bf{2}}},{{\bf{b}}_{\bf{3}}}} \right\}\) and \(D = \left\{ {{{\bf{d}}_{\bf{1}}},{{\bf{d}}_{\bf{2}}}} \right\}\) be bases for vector space \(V\) and \(W\), respectively. Let \(T:V \to W\) be a linear transformation with the property that

\(T\left( {{{\bf{b}}_1}} \right) = 3{{\bf{d}}_1} - 5{{\bf{d}}_2}\), \(T\left( {{{\bf{b}}_2}} \right) = - {{\bf{d}}_1} + 6{{\bf{d}}_2}\), \(T\left( {{{\bf{b}}_3}} \right) = 4{{\bf{d}}_2}\)

Find the matrix for \(T\) relative to \(B\), and\(D\).

Consider an invertiblen 脳 n matrix A such that the zero state is a stable equilibrium of the dynamical system x鈫�(t+1)=Ax鈫�(t)What can you say about the stability of the systems

x鈫�(t+1)=(A-2In)x鈫�(t)

Question: Exercises 9-14 require techniques section 3.1. Find the characteristic polynomial of each matrix, using either a cofactor expansion or the special formula for \(3 \times 3\) determinants described prior to Exercise 15-18 in Section 3.1. [Note: Finding the characteristic polynomial of a \(3 \times 3\) matrix is not easy to do with just row operations, because the variable \(\lambda \) is involved.

14. \(\left[ {\begin{array}{*{20}{c}}5&- 2&3\\0&1&0\\6&7&- 2\end{array}} \right]\)

For the matrix A, find real closed formulas for the trajectoryx鈫�(t+1)=Ax炉(t)where x鈫�=[01]. Draw a rough sketch

A=[15-27]
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