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Chapter 5: Q22SE (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 \(p\): \({C_p} = \left[ {\begin{aligned}{*{20}{c}}{\bf{0}}&{\bf{1}}&{\bf{0}}&{...}&{\bf{0}}\\{\bf{0}}&{\bf{0}}&{\bf{1}}&{}&{\bf{0}}\\:&{}&{}&{}&:\\{\bf{0}}&{\bf{0}}&{\bf{0}}&{}&{\bf{1}}\\{ - {a_{\bf{0}}}}&{ - {a_{\bf{1}}}}&{ - {a_{\bf{2}}}}&{...}&{ - {a_{n - {\bf{1}}}}}\end{aligned}} \right]\).

22. Let \(p\left( t \right) = {a_{\bf{0}}} + {a_{\bf{1}}}t + {a_{\bf{2}}}{t^{\bf{2}}} + {t^{\bf{3}}}\), and let \(\lambda \) be a zero of \(p\).

  1. Write the companion matrix for \(p\).
  2. Explain why \({\lambda ^{\bf{3}}} = - {a_{\bf{0}}} - {a_{\bf{1}}}\lambda - {a_{\bf{2}}}{\lambda ^{\bf{2}}}\), and show that \(\left( {{\bf{1}},\lambda ,{\lambda ^2}} \right)\) is an eigenvector of the companion matrix for \(p\).

Short Answer

Expert verified
  1. The companion matrix for \(p\)is\({C_p} = \left[ {\begin{aligned}{*{20}{c}}0&1&0\\0&0&1\\{ - {a_0}}&{ - {a_1}}&{ - {a_2}}\end{aligned}} \right]\).
  2. As \(\lambda \) is a zero of the matrix, this implies that \( - {a_0} - {a_1}\lambda - {a_2}{\lambda ^2} = {\lambda ^3}\). It is proved that \(\left( {1,\lambda ,{\lambda ^2}} \right)\) is an eigenvector of the companion matrix \(p\).

Step by step solution

01

Step 1: Write the companion matrix for \(p\)

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\({C_p} = \left[ {\begin{aligned}{*{20}{c}}0&1&0\\0&0&1\\{ - {a_0}}&{ - {a_1}}&{ - {a_2}}\end{aligned}} \right]\).

Thus, the companion matrix for \(p\)is\({C_p} = \left[ {\begin{aligned}{*{20}{c}}0&1&0\\0&0&1\\{ - {a_0}}&{ - {a_1}}&{ - {a_2}}\end{aligned}} \right]\).

02

Step 2: Show that \(\left( {{\bf{1}},\lambda ,{\lambda ^2}} \right)\) is an eigenvector of the companion matrix for \(p\)

As \(\lambda \) is a zero of the given polynomial \(p\left( t \right)\) then we have,

\(\begin{aligned}{c}{a_0} + {a_1}\lambda + {a_2}{\lambda ^2} + {\lambda ^3} = 0\\ - {a_0} - {a_1}\lambda - {a_2}{\lambda ^2} = {\lambda ^3}\end{aligned}\)

Now check whether \(\left( {1,\lambda ,{\lambda ^2}} \right)\) is an eigenvector of \({C_p}\).

\(\begin{aligned}{c}{C_p} &= \left[ {\begin{aligned}{*{20}{c}}1\\\lambda \\{{\lambda ^2}}\end{aligned}} \right]\\ &= \left[ {\begin{aligned}{*{20}{c}}0&1&0\\0&0&1\\{ - {a_0}}&{ - {a_1}}&{ - {a_2}}\end{aligned}} \right]\left[ {\begin{aligned}{*{20}{c}}1\\\lambda \\{{\lambda ^2}}\end{aligned}} \right]\\ &= \left[ {\begin{aligned}{*{20}{c}}\lambda \\{{\lambda ^2}}\\{ - {a_0} - {a_1}\lambda - {a_2}{\lambda ^2}}\end{aligned}} \right]\\ &= \left[ {\begin{aligned}{*{20}{c}}\lambda \\{{\lambda ^2}}\\{{\lambda ^3}}\end{aligned}} \right]\\ &= \lambda \left[ {\begin{aligned}{*{20}{c}}1\\\lambda \\{{\lambda ^2}}\end{aligned}} \right]\end{aligned}\)

Thus, \(\left( {1,\lambda ,{\lambda ^2}} \right)\) is an eigenvector of \({C_p}\).

Hence it is proved that \(\left( {1,\lambda ,{\lambda ^2}} \right)\) is an eigenvector of the companion matrix \(p\).

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Most popular questions from this chapter

A particle moving in a planar force field has a position vector .\(x\). that satisfies \(x' = Ax\). The \(2 \times 2\) matrix \(A\) has eigenvalues 4 and 2, with corresponding eigenvectors \({v_1} = \left( {\begin{aligned}{{20}{c}}{ - 3}\\1\end{aligned}} \right)\) and \({v_2} = \left( {\begin{aligned}{{20}{c}}{ - 1}\\1\end{aligned}} \right)\). Find the position of the particle at a time \(t\), assuming that \(x\left( 0 \right) = \left( {\begin{aligned}{{20}{c}}{ - 6}\\1\end{aligned}} \right)\).

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 \(p\): \({C_p} = \left( {\begin{aligned}{*{20}{c}}{\bf{0}}&{\bf{1}}&{\bf{0}}&{...}&{\bf{0}}\\{\bf{0}}&{\bf{0}}&{\bf{1}}&{}&{\bf{0}}\\:&{}&{}&{}&:\\{\bf{0}}&{\bf{0}}&{\bf{0}}&{}&{\bf{1}}\\{ - {a_{\bf{0}}}}&{ - {a_{\bf{1}}}}&{ - {a_{\bf{2}}}}&{...}&{ - {a_{n - {\bf{1}}}}}\end{aligned}} \right)\).

21. Use mathematical induction to prove that for \(n \ge {\bf{2}}\),\(\begin{aligned}{c}det\left( {{C_p} - \lambda I} \right) = {\left( { - {\bf{1}}} \right)^n}\left( {{a_{\bf{0}}} + {a_{\bf{1}}}\lambda + ... + {a_{n - {\bf{1}}}}{\lambda ^{n - {\bf{1}}}} + {\lambda ^n}} \right)\\ = {\left( { - {\bf{1}}} \right)^n}p\left( \lambda \right)\end{aligned}\)

(Hint: Expanding by cofactors down the first column, show that \(det\left( {{C_p} - \lambda I} \right)\) has the form \(\left( { - \lambda B} \right) + {\left( { - {\bf{1}}} \right)^n}{a_{\bf{0}}}\) where \(B\) is a certain polynomial (by the induction assumption).)

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

4. \(\left[ {\begin{array}{*{20}{c}}5&-3\\-4&3\end{array}} \right]\)

Let\(T:{{\rm P}_2} \to {{\rm P}_3}\) be a linear transformation that maps a polynomial \({\bf{p}}\left( t \right)\) into the polynomial \(\left( {t + 5} \right){\bf{p}}\left( t \right)\).

  1. Find the image of\({\bf{p}}\left( t \right) = 2 - t + {t^2}\).
  2. Show that \(T\) is a linear transformation.
  3. Find the matrix for \(T\) relative to the bases \(\left\{ {1,t,{t^2}} \right\}\) and \(\left\{ {1,t,{t^2},{t^3}} \right\}\).

Let\(\varepsilon = \left\{ {{{\bf{e}}_1},{{\bf{e}}_2},{{\bf{e}}_3}} \right\}\) be the standard basis for \({\mathbb{R}^3}\),\(B = \left\{ {{{\bf{b}}_1},{{\bf{b}}_2},{{\bf{b}}_3}} \right\}\) be a basis for a vector space \(V\) and\(T:{\mathbb{R}^3} \to V\) be a linear transformation with the property that

\(T\left( {{x_1},{x_2},{x_3}} \right) = \left( {{x_3} - {x_2}} \right){{\bf{b}}_1} - \left( {{x_1} - {x_3}} \right){{\bf{b}}_2} + \left( {{x_1} - {x_2}} \right){{\bf{b}}_3}\)

  1. Compute\(T\left( {{{\bf{e}}_1}} \right)\), \(T\left( {{{\bf{e}}_2}} \right)\) and \(T\left( {{{\bf{e}}_3}} \right)\).
  2. Compute \({\left( {T\left( {{{\bf{e}}_1}} \right)} \right)_B}\), \({\left( {T\left( {{{\bf{e}}_2}} \right)} \right)_B}\) and \({\left( {T\left( {{{\bf{e}}_3}} \right)} \right)_B}\).
  3. Find the matrix for \(T\) relative to \(\varepsilon \), and\(B\).
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