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Chapter 5: Q5.5-17E (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.

17. \(\left( {\begin{aligned}{}1&{}&{ - .8}\\4&{}&{ - 2.2}\end{aligned}} \right)\)

Short Answer

Expert verified

The invertible matrix \(P\)and matrix \(C\) are \(C = \left( {\begin{aligned}{}{ - .6}&{}&{ - .8}\\{.8}&{}&{ - .6}\end{aligned}} \right),P = \left( {\begin{aligned}{}2&{}&{ - 1}\\5&{}&0\end{aligned}} \right)\).

Step by step solution

01

Finding the matrix \(P\) and the matrix \(C\)

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 

Given that \(A = \left( {\begin{aligned}{}1&{}&{ - .8}\\4&{}&{ - 2.2}\end{aligned}} \right)\)

The characteristic equation is \({\lambda ^2} + 1.2\lambda + 1 = 0\).

Solving this we get \(\lambda = - 0.6 \pm 8i\).

To find an eigenvector corresponding to \( - .6 - .8i\) we compute,

\(A - \left( { - .6 - .8i} \right)I = \left( {\begin{aligned}{}{1.6 + .8i}&{}&{ - .8}\\4&{}&{ - 1.6 + .9i}\end{aligned}} \right)\)

The equation \(\left( {A - \left( { - .6 - .8i} \right)I} \right)v = 0\) gives the system of linear equations,

\(\begin{aligned}{}\left( {1.6 + 0.8i} \right)x - 0.8y &= 0\\4x + \left( { - 1.6 + .8i} \right)y &= 0\end{aligned}\)

Now from the 2nd equation, we get,

\(x = \left( {\left( {2 - i} \right)/5} \right)y\) with \(y\) is a free variable.

The eigenvector corresponds to \( - .6 - .8i\) is \(v = \left( {\begin{aligned}{}{2 - i}\\5\end{aligned}} \right)\).

By Theorem 9,

\(P = \left( {\begin{aligned}{}{{\rm{ Re v}}}&{}&{{\mathop{\rm Im}\nolimits} v}\end{aligned}} \right) \Rightarrow P = \left( {\begin{aligned}{}2&{}&{ - 1}\\5&{}&0\end{aligned}} \right)\)

03

Find the matrix  

After that, we can find the matrix by using\(C = {P^{ - 1}}AP\).

\(\begin{aligned}{}C {}&= {P^{ - 1}}AP\\ {}&= \frac{1}{5}\left( {\begin{aligned}{}0&{}&1\\{ - 5}&{}&1\end{aligned}} \right)\left( {\begin{aligned}{}1&{}&{ - .8}\\4&{ - 2.2}\end{aligned}} \right)\left( {\begin{aligned}{}2&{}&{ - 1}\\5&{}&0\end{aligned}} \right)\\ &= \left( {\begin{aligned}{}{ - .6}&{}&{ - .8}\\{.8}&{}&{ - .6}\end{aligned}} \right)\end{aligned}\)

Therefore, the matrixes are \(C = \left( {\begin{aligned}{}{ - .6}&{}&{ - .8}\\{.8}&{}&{ - .6}\end{aligned}} \right)\) and \(P = \left( {\begin{aligned}{}2&{}&{ - 1}\\5&{}&0\end{aligned}} \right)\).

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Most popular questions from this chapter

Let \({\bf{u}}\) be an eigenvector of \(A\) corresponding to an eigenvalue \(\lambda \), and let \(H\) be the line in \({\mathbb{R}^{\bf{n}}}\) through \({\bf{u}}\) and the origin.

  1. Explain why \(H\) is invariant under \(A\) in the sense that \(A{\bf{x}}\) is in \(H\) whenever \({\bf{x}}\) is in \(H\).
  2. Let \(K\) be a one-dimensional subspace of \({\mathbb{R}^{\bf{n}}}\) that is invariant under \(A\). Explain why \(K\) contains an eigenvector of \(A\).

Question: Is \(\left( {\begin{array}{*{20}{c}}1\\4\end{array}} \right)\) an eigenvalue of \(\left( {\begin{array}{*{20}{c}}{ - 3}&1\\{ - 3}&8\end{array}} \right)\)? If so, find the eigenvalue.

Question: Construct a random integer-valued \(4 \times 4\) matrix \(A\).

  1. Reduce \(A\) to echelon form \(U\) with no row scaling, and use \(U\) in formula (1) (before Example 2) to compute \(\det A\). (If \(A\) happens to be singular, start over with a new random matrix.)
  2. Compute the eigenvalues of \(A\) and the product of these eigenvalues (as accurately as possible).
  3. List the matrix \(A\), and, to four decimal places, list the pivots in \(U\) and the eigenvalues of \(A\). Compute \(\det A\) with your matrix program, and compare it with the products you found in (a) and (b).

Let \(J\) be the \(n \times n\) matrix of all \({\bf{1}}\)鈥檚 and consider \(A = \left( {a - b} \right)I + bJ\) that is,

\(A = \left( {\begin{aligned}{*{20}{c}}a&b&b&{...}&b\\b&a&b&{...}&b\\b&b&a&{...}&b\\:&:&:&:&:\\b&b&b&{...}&a\end{aligned}} \right)\)

Use the results of Exercise \({\bf{16}}\) in the Supplementary Exercises for Chapter \({\bf{3}}\) to show that the eigenvalues of \(A\) are \(a - b\) and \(a + \left( {n - {\bf{1}}} \right)b\). What are the multiplicities of these eigenvalues?

Question: Is \(\left( {\begin{array}{*{20}{c}}{ - 1 + \sqrt 2 }\\1\end{array}} \right)\) an eigenvalue of \(\left( {\begin{array}{*{20}{c}}2&1\\1&4\end{array}} \right)\)? If so, find the eigenvalue.
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