Chapter 5: Eigenvalues and Eigenvectors

In Exercises 9–14, classify the origin as an attractor, repeller, or saddle point of the dynamical system ${\mathrm{x}}_{k+1}=A{\mathrm{x}}_k$. Find the directions of greatest attraction and/or repulsion.

9. $A=\left(\begin{array}{rlr}1.7& & -.3\\ -1.2& & .8\end{array}\right)$

Let $A=\left(\begin{array}{rl}20r15& 16\\ -20& -21\end{array}\right)$. The vectors $\mathbf{x},\dots ,A^5\mathbf{x}$ are $\left(\begin{array}{r}20l1\\ 1\end{array}\right)$,

$\left(\begin{array}{r}20r31\\ -41\end{array}\right),\left(\begin{array}{r}20r-191\\ 241\end{array}\right),\left(\begin{array}{r}20r991\\ -1241\end{array}\right),\left(\begin{array}{r}20r-4991\\ 6241\end{array}\right),\left(\begin{array}{r}20r24991\\ -31241\end{array}\right)$.

Find a vector with a 1 in the second entry that is close to an eigenvector of $A$. Use four decimal places. Check your estimate, and give an estimate for the dominant eigenvalue of $A$.

In Exercises 3-6, solve the initial value problem $x^{\prime }\left(t\right)=Ax\left(t\right)$ 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 $x^{\prime }=Ax$. Find the directions of greatest attraction and/or repulsion. When the origin is a saddle point, sketch typical trajectories.

5. $A=\left(\begin{array}{rl}20c7& -1\\ 3& 3\end{array}\right)$

Question: Is $\left(\begin{array}{c}4\\ -3\\ 1\end{array}\right)$ an eigenvector of $\left(\begin{array}{ccc}3& 7& 9\\ -4& -5& 1\\ 2& 4& 4\end{array}\right)$? If so, find the eigenvalue.

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$p\left(\lambda \right)$.

In Exercises 3-6, solve the initial value problem $x^{\prime }\left(t\right)=Ax\left(t\right)$ 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 $x^{\prime }=Ax$. Find the directions of greatest attraction and/or repulsion. When the origin is a saddle point, sketch typical trajectories.

6. $A=\left(\begin{array}{rl}20c1& -2\\ 3& -4\end{array}\right)$

Question: Is $\left(\begin{array}{c}1\\ -2\\ 1\end{array}\right)$ an eigenvector of$\left)\begin{array}{ccc}3& 6& 7\\ 3& 3& 7\\ 5& 6& 5\end{array}\right)$? If so, find the eigenvalue.

Let $A=\left(\begin{array}{rl}20r-2& -3\\ 6& 7\end{array}\right)$. Repeat Exercise 5, using the following sequence $\mathbf{x},A\mathbf{x},\dots ,A^5\mathbf{x}$.

$\left(\begin{array}{r}20l1\\ 1\end{array}\right),\left(\begin{array}{r}20r-5\\ 13\end{array}\right),\left(\begin{array}{r}20r-29\\ 61\end{array}\right),\left(\begin{array}{r}20r-125\\ 253\end{array}\right),\left(\begin{array}{r}20r-509\\ 1021\end{array}\right),\left(\begin{array}{r}20r-2045\\ 4093\end{array}\right)$

Let

${\mathbf{v}}_{\mathbf{1}}=\left(\begin{array}{r}20c\mathbf{2}\\ \mathbf{0}\\ -\mathbf{1}\\ \mathbf{2}\end{array}\right)$, ${\mathbf{v}}_{\mathbf{2}}=\left(\begin{array}{r}20c\mathbf{0}\\ -\mathbf{2}\\ \mathbf{2}\\ \mathbf{1}\end{array}\right)$, and let ${\mathbf{v}}_{\mathbf{3}}=\left(\begin{array}{r}20c-\mathbf{2}\\ \mathbf{1}\\ \mathbf{0}\\ \mathbf{2}\end{array}\right)$be the orthogonal set $\left\{{\mathbf{v}}_{\mathbf{1}},{\mathbf{v}}_{\mathbf{2}},{\mathbf{v}}_{\mathbf{3}},{\mathbf{v}}_{\mathbf{4}}\right\}$. Determine whether ${\mathbf{p}}_i$ is in span S, $\mathbf{a}\mathbf{f}\mathbf{f}S$, or $\mathbf{c}\mathbf{o}\mathbf{n}\mathbf{v}S$.

a. ${\mathbf{p}}_{\mathbf{1}}$ b. ${\mathbf{p}}_{\mathbf{2}}$ c. ${\mathbf{p}}_{\mathbf{3}}$ d. ${\mathbf{p}}_{\mathbf{4}}$

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

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.