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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.)
It is proved that \(\lambda \) must be the eigenvalue of both A and \({A^T}\).
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In Exercises 3-6, solve the initial value problem \(x'\left( t \right) = Ax\left( t \right)\) for \(t \ge 0\), with \(x\left( 0 \right) = \left( {3,2} \right)\). Classify the nature of the origin as an attractor, repeller, or saddle point of the dynamical system described by \(x' = Ax\). Find the directions of greatest attraction and/or repulsion. When the origin is a saddle point, sketch typical trajectories.
5. \(A = \left( {\begin{aligned}{ {20}{c}}7&{ - 1}\\3&3\end{aligned}} \right)\)
Question: Find the characteristic polynomial and the eigenvalues of the matrices in Exercises 1-8.
7. \(\left[ {\begin{array}{*{20}{c}}5&3\\- 4&4\end{array}} \right]\)
In Exercises \({\bf{3}}\) and \({\bf{4}}\), use the factorization \(A = PD{P^{ - {\bf{1}}}}\) to compute \({A^k}\) where \(k\) represents an arbitrary positive integer.
4. \(\left( {\begin{array}{*{20}{c}}{ - 2}&{12}\\{ - 1}&5\end{array}} \right) = \left( {\begin{array}{*{20}{c}}3&4\\1&1\end{array}} \right)\left( {\begin{array}{*{20}{c}}2&0\\0&1\end{array}} \right)\left( {\begin{array}{*{20}{c}}{ - 1}&4\\1&{ - 3}\end{array}} \right)\)
Question: Exercises 9-14 require techniques section 3.1. Find the characteristic polynomial of each matrix, using either a cofactor expansion or the special formula for \(3 \times 3\) determinants described prior to Exercise 15-18 in Section 3.1. [Note: Finding the characteristic polynomial of a \(3 \times 3\) matrix is not easy to do with just row operations, because the variable \(\lambda \) is involved.
13. \(\left[ {\begin{array}{*{20}{c}}6&- 2&0\\- 2&9&0\\5&8&3\end{array}} \right]\)
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.
6. \(\left( {\begin{array}{*{20}{c}}{\bf{4}}&{\bf{0}}&{{\bf{ - 2}}}\\{\bf{2}}&{\bf{5}}&{\bf{4}}\\{\bf{0}}&{\bf{0}}&{\bf{5}}\end{array}} \right){\bf{ = }}\left( {\begin{array}{*{20}{c}}{{\bf{ - 2}}}&{\bf{0}}&{{\bf{ - 1}}}\\{\bf{0}}&{\bf{1}}&{\bf{2}}\\{\bf{1}}&{\bf{0}}&{\bf{0}}\end{array}} \right)\left( {\begin{array}{*{20}{c}}{\bf{5}}&{\bf{0}}&{\bf{0}}\\{\bf{0}}&{\bf{5}}&{\bf{0}}\\{\bf{0}}&{\bf{0}}&4\end{array}} \right)\left( {\begin{array}{*{20}{c}}{\bf{0}}&{\bf{0}}&{\bf{1}}\\{\bf{2}}&{\bf{1}}&{\bf{4}}\\{{\bf{ - 1}}}&{\bf{0}}&{{\bf{ - 2}}}\end{array}} \right)\)
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