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Chapter 5: Q37E (page 267)

[M] In Exercises 37鈥�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

Suppose \(A = PD{P^{ - 1}}\), where \(P\) is \(2 \times 2\) and \(D = \left( {\begin{array}{*{20}{l}}2&0\\0&7\end{array}} \right)\)

a. Let \(B = 5I - 3A + {A^2}\). Show that \(B\) is diagonalizable by finding a suitable factorization of \(B\).

b. Given \(p\left( t \right)\) and \(p\left( A \right)\) as in Exercise 5 , show that \(p\left( A \right)\) is diagonalizable.

For the matrix A, find real closed formulas for the trajectory x鈫�(t+1)=Ax炉(t) where x鈫�=[01]. Draw a rough sketchA=[7-156-11]

(M)The MATLAB command roots\(\left( p \right)\) computes the roots of the polynomial equation \(p\left( t \right) = {\bf{0}}\). Read a MATLAB manual, and then describe the basic idea behind the algorithm for the roots command.

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?

If \(p\left( t \right) = {c_0} + {c_1}t + {c_2}{t^2} + ...... + {c_n}{t^n}\), define \(p\left( A \right)\) to be the matrix formed by replacing each power of \(t\) in \(p\left( t \right)\)by the corresponding power of \(A\) (with \({A^0} = I\) ). That is,

\(p\left( t \right) = {c_0} + {c_1}I + {c_2}{I^2} + ...... + {c_n}{I^n}\)

Show that if \(\lambda \) is an eigenvalue of A, then one eigenvalue of \(p\left( A \right)\) is\(p\left( \lambda \right)\).
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