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Chapter 5: Q27E (page 267)

Question: Show that if \(\lambda \) is an eigenvalue of A if and only if \(\lambda \) is an eigenvalue of \({A^T}\). (Find out how \(A - \lambda I\) and \({A^T} - \lambda I\) are related.)

Short Answer

Expert verified

It is proved that \(\lambda \) must be the eigenvalue of both A and \({A^T}\).

Step by step solution

01

Write the given information

Here, \(\lambda \) is the eigenvalue of a matrix A.

02

Check for the eigenvalue of \(A\)

Find the relation between \(A - \lambda I\) and \({A^T} - \lambda I\).

\(\begin{array}{c}{\left( {A - \lambda I} \right)^T} = {A^T} - {\left( {\lambda I} \right)^T}\\ = {A^T} - \lambda I\end{array}\)

So, \({\left( {A - \lambda I} \right)^T}\) is invertible if and only if \(A - \lambda I\) is invertible.

According to theorem 6(c), \({A^T} - \lambda I\) is not invertible if only if \(A - \lambda I\) is not invertible.

Therefore, \(\lambda \) must be the eigenvalue of both A and \({A^T}\).

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Most popular questions from this chapter

(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.

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

6. \(\left[ {\begin{array}{*{20}{c}}3&- 4\\4&8\end{array}} \right]\)

Question: In Exercises \({\bf{5}}\) and \({\bf{6}}\), the matrix \(A\) is factored in the form \(PD{P^{ - {\bf{1}}}}\). Use the Diagonalization Theorem to find the eigenvalues of \(A\) and a basis for each eigenspace.

5. \(\left( {\begin{array}{*{20}{c}}2&2&1\\1&3&1\\1&2&2\end{array}} \right) = \left( {\begin{array}{*{20}{c}}1&1&2\\1&0&{ - 1}\\1&{ - 1}&0\end{array}} \right)\left( {\begin{array}{*{20}{c}}5&0&0\\0&1&0\\0&0&1\end{array}} \right)\left( {\begin{array}{*{20}{c}}{\frac{1}{4}}&{\frac{1}{2}}&{\frac{1}{4}}\\{\frac{1}{4}}&{\frac{1}{2}}&{ - \frac{3}{4}}\\{\frac{1}{4}}&{ - \frac{1}{2}}&{\frac{1}{4}}\end{array}} \right)\)

(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.

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 \(\left( {{\bf{ - 2,}}\;{\bf{1,}}\;{\bf{2}}} \right)\).

8. \(\left( {\begin{array}{*{20}{c}}{\bf{5}}&{\bf{1}}\\{\bf{0}}&{\bf{5}}\end{array}} \right)\)
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