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Chapter 5: Q2E (page 267)

In Exercises 1鈥�4, the matrix A is followed by a sequence \(\left\{ {{{\bf{x}}_k}} \right\}\] produced by the power method. Use these data to estimate the largest eigenvalue of A, and give a corresponding eigenvector.

2. \(A = \left( {\begin{aligned}{ {20}{c}}{1.8}&{ - .8}\\{ - 3.2}&{4.2}\end{aligned}} \right]\)

\(\left( {\begin{aligned}{ {20}{c}}1\\0\end{aligned}} \right],{\rm{ }}\left( {\begin{aligned}{ {20}{c}}{ - .5625}\\1\end{aligned}} \right],{\rm{ }}\left( {\begin{aligned}{ {20}{c}}{ - .3021}\\1\end{aligned}} \right],{\rm{ }}\left( {\begin{aligned}{ {20}{c}}{ - .2601}\\1\end{aligned}} \right],{\rm{ }}\left( {\begin{aligned}{ {20}{c}}{ - .2520}\\1\end{aligned}} \right]\)

Short Answer

Expert verified

The largest eigenvalue of A is 5.0064, and the corresponding eigenvector is \(\left( {\begin{aligned}{ {20}{c}}{ - .2560}\\1\end{aligned}} \right]\).

Step by step solution

01

Given information

A matrix \(A = \left( {\begin{aligned}{ {20}{l}}{1.8}&{ - .8}\\{ - 3.2}&{4.2}\end{aligned}} \right]\). A sequence \(\left\{ {{x_k}} \right\}\).

02

Find the Eigenvalue

Compute the value of\(A{x_k}\)and identify the largest entry as follows:

\(A{x_0} = \left( {\begin{aligned}{ {20}{c}}{1.8}&{ - .8}\\{ - 3.2}&{4.2}\end{aligned}} \right]\left( {\begin{aligned}{ {20}{l}}1\\0\end{aligned}} \right] = \left( {\begin{aligned}{ {20}{c}}{1.8}\\{ - 3.2}\end{aligned}} \right]\),\({\mu _0} = - 3.2\)

\(A{x_1} = \left( {\begin{aligned}{ {20}{c}}{1.8}&{ - .8}\\{ - 3.2}&{4.2}\end{aligned}} \right]\left( {\begin{aligned}{ {20}{c}}{ - .5625}\\1\end{aligned}} \right] = \left( {\begin{aligned}{ {20}{c}}{ - 1.8125}\\{4.7625}\end{aligned}} \right]\),\({\mu _1} = 4.7625\)

\(A{x_2} = \left( {\begin{aligned}{ {20}{c}}{1.8}&{ - .8}\\{ - 3.2}&{4.2}\end{aligned}} \right]\left( {\begin{aligned}{ {20}{c}}{ - .3021}\\1\end{aligned}} \right] = \left( {\begin{aligned}{ {20}{c}}{ - 1.34378}\\{5.16672}\end{aligned}} \right]\),\({\mu _2} = 5.16672\)

\(A{x_3} = \left( {\begin{aligned}{ {20}{c}}{1.8}&{ - .8}\\{ - 3.2}&{4.2}\end{aligned}} \right]\left( {\begin{aligned}{ {20}{c}}{ - .2601}\\1\end{aligned}} \right] = \left( {\begin{aligned}{ {20}{c}}{ - 1.26818}\\{5.03232}\end{aligned}} \right]\),\({\mu _3} = 5.03232\)

\(A{x_3} = \left( {\begin{aligned}{ {20}{c}}{1.8}&{ - .8}\\{ - 3.2}&{4.2}\end{aligned}} \right]\left( {\begin{aligned}{ {20}{c}}{ - .2520}\\1\end{aligned}} \right] = \left( {\begin{aligned}{ {20}{c}}{ - 1.2536}\\{5.0064}\end{aligned}} \right]\),\({\mu _4} = 5.0064\)

Hence, the largest absolute entry is 5.0064. So, the eigenvalue is equal to 4.9978. The corresponding eigenvector is \(\left( {\begin{aligned}{ {20}{c}}{ - .2520}\\1\end{aligned}} \right]\).

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

A common misconception is that if \(A\) has a strictly dominant eigenvalue, then, for any sufficiently large value of \(k\), the vector \({A^k}{\bf{x}}\) is approximately equal to an eigenvector of \(A\). For the three matrices below, study what happens to \({A^k}{\bf{x}}\) when \({\bf{x = }}\left( {{\bf{.5,}}{\bf{.5}}} \right)\), and try to draw general conclusions (for a \({\bf{2 \times 2}}\) matrix).

a. \(A{\bf{ = }}\left( {\begin{aligned}{ {20}{c}}{{\bf{.8}}}&{\bf{0}}\\{\bf{0}}&{{\bf{.2}}}\end{aligned}} \right)\) b. \(A{\bf{ = }}\left( {\begin{aligned}{ {20}{c}}{\bf{1}}&{\bf{0}}\\{\bf{0}}&{{\bf{.8}}}\end{aligned}} \right)\) c. \(A{\bf{ = }}\left( {\begin{aligned}{ {20}{c}}{\bf{8}}&{\bf{0}}\\{\bf{0}}&{\bf{2}}\end{aligned}} \right)\)

Let \(A{\bf{ = }}\left( {\begin{aligned}{*{20}{c}}{{a_{{\bf{11}}}}}&{{a_{{\bf{12}}}}}\\{{a_{{\bf{21}}}}}&{{a_{{\bf{22}}}}}\end{aligned}} \right)\). Recall from Exercise \({\bf{25}}\) in Section \({\bf{5}}{\bf{.4}}\) that \({\rm{tr}}\;A\) (the trace of \(A\)) is the sum of the diagonal entries in \(A\). Show that the characteristic polynomial of \(A\) is \({\lambda ^2} - \left( {{\rm{tr}}A} \right)\lambda + \det A\). Then show that the eigenvalues of a \({\bf{2 \times 2}}\) matrix \(A\) are both real if and only if \(\det A \le {\left( {\frac{{{\rm{tr}}A}}{2}} \right)^2}\).

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

5. \(\left[ {\begin{array}{*{20}{c}}2&1\\-1&4\end{array}} \right]\)

Consider an invertible n 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-1x鈫�(t)
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