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Chapter 5: Q5.5-15E (page 267)

In Exercises 13-20, find an invertible matrix \(P\) and a matrix \(C\) of the form \(\left( {\begin{aligned}{}a&{}&{ - b}\\b&{}&a\end{aligned}} \right)\) such that the given matrix has the form \(A = PC{P^{ - 1}}\). For Exercises 13-16, use information from Exercises 1-4.

15.\(\left( {\begin{aligned}{}1&{}&5\\{ - 2}&{}&3\end{aligned}} \right)\)

Short Answer

Expert verified

The invertible matrix \(P\) and matrix \(C\) are \(P = \left( {\begin{aligned}{}1&{}&3\\2&{}&0\end{aligned}} \right),C = \left( {\begin{aligned}{}2&{}&{ - 3}\\3&{}&2\end{aligned}} \right)\).

Step by step solution

01

Finding the matrix \(P\) and the matrix \(C\)  such that  \(A = PC{P^{ - 1}}\)

Let \(A\) be a \(2 \times 2\) with eigenvalues of the form \(a \pm bi\) , and let \(v\) be the eigenvector corresponding to the eigenvalue \(a - bi\), then the matrix \(P\) will be \(P = \left( {\begin{aligned}{}{{\mathop{\rm Re}\nolimits} v}&{{\mathop{\rm Im}\nolimits} v}\end{aligned}} \right)\).

The matrix \(C\) can be obtained by \(C = {P^{ - 1}}AP\).

02

Find the Invertible matrix

From Exercise 3, the eigenvalues of\(A\)are\(\lambda = 2 \pm 3i\), and \(v = \left( {\begin{aligned}{}{1 + 3i}\\2\end{aligned}} \right)\) be the eigenvector correspond to the eigenvalue \(\lambda = 2 - 3i\).

By theorem 9,

\(\begin{aligned}{}P &= \left( {\begin{aligned}{}{{\mathop{\rm Re}\nolimits} v}&{}&{{\mathop{\rm Im}\nolimits} v}\end{aligned}} \right)\\P &= \left( {\begin{aligned}{}1&{}&3\\2&{}&0\end{aligned}} \right)\end{aligned}\)

Substitute the value of\({P^{ - 1}}\), \(P\) and \(A\) into \(C = {P^{ - 1}}AP\) to obtain matrix \(C\).

\(\begin{aligned}{}C &= {P^{ - 1}}AP\\C &= \frac{1}{6}\left( {\begin{aligned}{}0&{}&{ - 3}\\{ - 2}&{}&1\end{aligned}} \right)\left( {\begin{aligned}{}1&{}&5\\{ - 2}&{}&3\end{aligned}} \right)\left( {\begin{aligned}{}1&{}&3\\2&{}&0\end{aligned}} \right)\\C &= \left( {\begin{aligned}{}2&{}&{ - 3}\\3&{}&2\end{aligned}} \right)\end{aligned}\)

Hence \(P = \left( {\begin{aligned}{}1&{}&3\\2&{}&0\end{aligned}} \right)\) and \(C = \left( {\begin{aligned}{}2&{}&{ - 3}\\3&{}&2\end{aligned}} \right)\).

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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\(D\).

Question: Let \(A = \left( {\begin{array}{*{20}{c}}{.6}&{.3}\\{.4}&{.7}\end{array}} \right)\), \({v_1} = \left( {\begin{array}{*{20}{c}}{3/7}\\{4/7}\end{array}} \right)\), \({x_0} = \left( {\begin{array}{*{20}{c}}{.5}\\{.5}\end{array}} \right)\). (Note: \(A\) is the stochastic matrix studied in Example 5 of Section 4.9.)

  1. Find a basic for \({\mathbb{R}^2}\) consisting of \({{\rm{v}}_1}\) and anther eigenvector \({{\rm{v}}_2}\) of \(A\).
  2. Verify that \({{\rm{x}}_0}\) may be written in the form \({{\rm{x}}_0} = {{\rm{v}}_1} + c{{\rm{v}}_2}\).
  3. For \(k = 1,2, \ldots \), define \({x_k} = {A^k}{x_0}\). Compute \({x_1}\) and \({x_2}\), and write a formula for \({x_k}\). Then show that \({{\bf{x}}_k} \to {{\bf{v}}_1}\) as \(k\) increases.

Question: Diagonalize the matrices in Exercises \({\bf{7--20}}\), if possible. The eigenvalues for Exercises \({\bf{11--16}}\) are as follows:\(\left( {{\bf{11}}} \right)\lambda {\bf{ = 1,2,3}}\); \(\left( {{\bf{12}}} \right)\lambda {\bf{ = 2,8}}\); \(\left( {{\bf{13}}} \right)\lambda {\bf{ = 5,1}}\); \(\left( {{\bf{14}}} \right)\lambda {\bf{ = 5,4}}\); \(\left( {{\bf{15}}} \right)\lambda {\bf{ = 3,1}}\); \(\left( {{\bf{16}}} \right)\lambda {\bf{ = 2,1}}\). For exercise \({\bf{18}}\), one eigenvalue is \(\lambda {\bf{ = 5}}\) and one eigenvector is \(\left( {{\bf{ - 2,}}\;{\bf{1,}}\;{\bf{2}}} \right)\).

10. \(\left( {\begin{array}{*{20}{c}}{\bf{2}}&{\bf{3}}\\{\bf{4}}&{\bf{1}}\end{array}} \right)\)

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

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

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