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Chapter 5: Q37E (page 267)

[M] In Exercises 37–40, use a matrix program to find the eigenvalues of the matrix. Then use the method of Example 4 with a row reduction routine to produce a basis for each eigenspace.

37. \(\left( {\begin{array}{*{20}{c}}8&{ - 10}&{ - 5}\\2&{17}&2\\{ - 9}&{ - 18}&4\end{array}} \right)\)

Short Answer

Expert verified

The eigenvalues are \(\lambda = \left( {3,13,13} \right)\). The basis for eigenspace of \(\lambda = 3\) is \(N = \left( {\begin{array}{*{20}{c}}5\\{ - 2}\\9\end{array}} \right)\). The basis for eigenspace of \(\lambda = 13\) is \(N = \left\{ {\left( {\begin{array}{*{20}{c}}{ - 2}\\1\\0\end{array}} \right),\left( {\begin{array}{*{20}{c}}{ - 1}\\0\\1\end{array}} \right)} \right\}\).

Step by step solution

01

Command for input the matrix in MATLAB

Write the command given below to input the matrix

\( > > {\rm{ A }} = {\rm{ }}\left( {{\rm{8 }} - 10{\rm{ }} - {\rm{5}};{\rm{ 2 17 2}};{\rm{ }} - {\rm{9 }} - 18{\rm{ 4}}} \right)\)

02

Command for finding the eigenvalues of the matrix

\( > > {\rm{ ev}} = {\rm{eig}}\left( {\rm{A}} \right)\)

The output will be\({\rm{ev}} = \left( {3,13,13} \right)\).

Hence, the eigenvalues are \(\lambda = \left( {3,13,13} \right)\).

03

Command for finding the null basis for each eigenvalue

Write the command given below to find the null basis corresponding to \(\lambda = 3\):

\( > > {\rm{ N}} = {\rm{A}} - {\rm{ev}}\left( 1 \right)*{\rm{eye}}\left( 3 \right)\)

The output is given below:

Hence, the basis for eigenspace of \(\lambda = 3\) is \(N = \left( {\begin{array}{*{20}{c}}5\\{ - 2}\\9\end{array}} \right)\).

Write the command given below to find the null basis corresponding to \(\lambda = 3\):

\( > > {\rm{ N}} = {\rm{A}} - {\rm{ev}}\left( 2 \right)*{\rm{eye}}\left( 3 \right)\)

The output is given below:

\(N = \left( {\begin{array}{*{20}{c}}{ - 2}&{ - 1}\\1&0\\0&1\end{array}} \right)\)

Hence, the basis for eigenspace of \(\lambda = 13\) is \(N = \left\{ {\left( {\begin{array}{*{20}{c}}{ - 2}\\1\\0\end{array}} \right),\left( {\begin{array}{*{20}{c}}{ - 1}\\0\\1\end{array}} \right)} \right\}\).

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Most popular questions from this chapter

Question: Let \(A = \left( {\begin{array}{*{20}{c}}{.5}&{.2}&{.3}\\{.3}&{.8}&{.3}\\{.2}&0&{.4}\end{array}} \right)\), \({{\rm{v}}_1} = \left( {\begin{array}{*{20}{c}}{.3}\\{.6}\\{.1}\end{array}} \right)\), \({{\rm{v}}_2} = \left( {\begin{array}{*{20}{c}}1\\{ - 3}\\2\end{array}} \right)\), \({{\rm{v}}_3} = \left( {\begin{array}{*{20}{c}}{ - 1}\\0\\1\end{array}} \right)\) and \({\rm{w}} = \left( {\begin{array}{*{20}{c}}1\\1\\1\end{array}} \right)\).

  1. Show that \({{\rm{v}}_1}\), \({{\rm{v}}_2}\), and \({{\rm{v}}_3}\) are eigenvectors of \(A\). (Note: \(A\) is the stochastic matrix studied in Example 3 of Section 4.9.)
  2. Let \({{\rm{x}}_0}\) be any vector in \({\mathbb{R}^3}\) with non-negative entries whose sum is 1. (In section 4.9, \({{\rm{x}}_0}\) was called a probability vector.) Explain why there are constants \({c_1}\), \({c_2}\), and \({c_3}\) such that \({{\rm{x}}_0} = {c_1}{{\rm{v}}_1} + {c_2}{{\rm{v}}_2} + {c_3}{{\rm{v}}_3}\). Compute \({{\rm{w}}^T}{{\rm{x}}_0}\), and deduce that \({c_1} = 1\).
  3. For \(k = 1,2, \ldots ,\) define \({{\rm{x}}_k} = {A^k}{{\rm{x}}_0}\), with \({{\rm{x}}_0}\) as in part (b). Show that \({{\rm{x}}_k} \to {{\rm{v}}_1}\) as \(k\) increases.

Question 19: Let \(A\) be an \(n \times n\) matrix, and suppose A has \(n\) real eigenvalues, \({\lambda _1},...,{\lambda _n}\), repeated according to multiplicities, so that \(\det \left( {A - \lambda I} \right) = \left( {{\lambda _1} - \lambda } \right)\left( {{\lambda _2} - \lambda } \right) \ldots \left( {{\lambda _n} - \lambda } \right)\) . Explain why \(\det A\) is the product of the n eigenvalues of A. (This result is true for any square matrix when complex eigenvalues are considered.)

Question: Construct a random integer-valued \(4 \times 4\) matrix \(A\), and verify that \(A\) and \({A^T}\) have the same characteristic polynomial (the same eigenvalues with the same multiplicities). Do \(A\) and \({A^T}\) have the same eigenvectors? Make the same analysis of a \(5 \times 5\) matrix. Report the matrices and your conclusions.

(M)Use a matrix program to diagonalize

\(A = \left( {\begin{aligned}{*{20}{c}}{ - 3}&{ - 2}&0\\{14}&7&{ - 1}\\{ - 6}&{ - 3}&1\end{aligned}} \right)\)

If possible. Use the eigenvalue command to create the diagonal matrix \(D\). If the program has a command that produces eigenvectors, use it to create an invertible matrix \(P\). Then compute \(AP - PD\) and \(PD{P^{{\bf{ - 1}}}}\). Discuss your results.

For the Matrices A find real closed formulas for the trajectory x→(t+1)=Ax→(t)where x(0)→=[01]

A=[43-34]
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