91Ó°ÊÓ

Find all complex eigenvalues of the matrices in Exercises 20 through 26 (including the real ones, of course). Do not use technology. Show all your work.

$\left[\begin{array}{ccc}1& 11& 1\\ 1& 11& 1\\ 0& 01& 1\\ 0& 01& 1\end{array}\right]$

TRUE OR FALSE

If two n x nmatrices A and B are diagonalizable, then AB must be diagonalizable as well.

26: Based on your answers in Exercises 24 and 25, sketch a phase portrait of the dynamical system

_{$\stackrel{\mathbf{¯}}{\mathbf{x}}\left(t+1\right)\mathbf{=}\left[\begin{array}{cc}0.5& 0.25\\ 0.5& 0.75\end{array}\right]\stackrel{\mathbf{¯}}{\mathbf{x}}\mathbf{\left(}\mathit{t}\mathbf{\right)}$}

Show that if a 6 × 6 matrix A has a negative determinant, then A has at least one positive eigenvalue. Hint: Sketch the graph of the characteristic polynomial

Consider a dynamical system $\stackrel{\mathbf{→}}{\mathbf{x}}\left(t+1\right)\mathbf{=}\mathit{A}\stackrel{\mathbf{→}}{\mathbf{x}}\mathbf{\left(}\mathit{t}\mathbf{\right)}$with two components. The accompanying sketch shows the initial state vectorlocalid="1668400938891" ${\stackrel{\mathbf{→}}{\mathbf{x}}}_{\mathbf{0}}$and two eigen vectorslocalid="1668400903162" $\stackrel{\mathbf{→}}{{\mathbf{Ï\dots }}_{\mathbf{1}}}\mathbf{ }\mathbf{ }\mathit{a}\mathit{n}\mathit{d}\mathbf{ }\mathbf{ }\stackrel{\mathbf{→}}{{\mathbf{Ï\dots }}_{\mathbf{2}}}$of A (with eigen valueslocalid="1668400914784" $\stackrel{\mathbf{→}}{{\mathbf{λ}}_{\mathbf{1}}}\mathbf{}\mathbf{\text{and}}\mathbf{}\stackrel{\mathbf{→}}{{\mathbf{λ}}_{\mathbf{2}}}$respectively). For the given values oflocalid="1668400926197" $\stackrel{\mathbf{→}}{{\mathbf{λ}}_{\mathbf{1}}}\mathbf{}\mathbf{\text{and}}\mathbf{}\stackrel{\mathbf{→}}{{\mathbf{λ}}_{\mathbf{2}}}$, draw a rough trajectory. Consider the future and the past of the system.

$\stackrel{\mathbf{→}}{{\mathbf{λ}}_{\mathbf{1}}}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{1}\mathbf{,}\stackrel{\mathbf{→}}{{\mathbf{λ}}_{\mathbf{2}}}\mathbf{=}\mathbf{1}$

For the matrices A and the vectors$\stackrel{\mathbf{→}}{{\mathbf{x}}_{\mathbf{0}}}$in Exercises 25 through 29, find $\underset{\mathbf{t}\mathbf{→}\mathbf{∞}}{\mathbf{lim}}\left(A^t{\stackrel{→}{x}}_0\right)$. Feel free to use Theorem 7.4.1.

$\mathbf{A}\mathbf{=}\mathbf{\left[}\begin{array}{cc}\mathbf{0}\mathbf{.}\mathbf{4}& \mathbf{0}\mathbf{.}\mathbf{5}\\ \mathbf{0}\mathbf{.}\mathbf{6}& \mathbf{0}\mathbf{.}\mathbf{5}\end{array}\mathbf{\right]}\mathbf{,}\stackrel{\mathbf{→}}{{\mathbf{x}}_{\mathbf{0}}}\mathbf{=}\mathbf{\left[}\begin{array}{c}\mathbf{0}\mathbf{.}\mathbf{54}\\ \mathbf{0}\mathbf{.}\mathbf{46}\end{array}\mathbf{\right]}$

Consider a dynamical system $\stackrel{\mathbf{→}}{\mathbf{x}}\left(t+1\right)\mathbf{=}\mathit{A}\stackrel{\mathbf{→}}{\mathbf{x}}\mathbf{\left(}\mathit{t}\mathbf{\right)}$with two components. The accompanying sketch shows the initial state vector ${\stackrel{\mathbf{→}}{\mathbf{x}}}_{\mathbf{0}}$and two eigen vectors $\stackrel{\mathbf{→}}{{\mathbf{Ï\dots }}_{\mathbf{1}}}\mathbf{ }\mathbf{ }\mathit{a}\mathit{n}\mathit{d}\mathbf{ }\mathbf{ }\stackrel{\mathbf{→}}{{\mathbf{Ï\dots }}_{\mathbf{2}}}$of A (with eigen values $\stackrel{\mathbf{→}}{{\mathbf{λ}}_{\mathbf{1}}}\mathbf{}\mathbf{\text{and}}\mathbf{}\stackrel{\mathbf{→}}{{\mathbf{λ}}_{\mathbf{2}}}$respectively). For the given values of $\stackrel{\mathbf{→}}{{\mathbf{λ}}_{\mathbf{1}}}\mathbf{}\mathbf{\text{and}}\mathbf{}\stackrel{\mathbf{→}}{{\mathbf{λ}}_{\mathbf{2}}}$, draw a rough trajectory. Consider the future and the past of the system.

$\stackrel{\mathbf{→}}{{\mathbf{λ}}_{\mathbf{1}}}\mathbf{=}\mathbf{1}\mathbf{,}\stackrel{\mathbf{→}}{{\mathbf{λ}}_{\mathbf{2}}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{9}$

If an invertible matrix A is diagonalizable, then A^-1 must be diagonalizable as well.

27: a. Based on your answers in Exercises 24 and 25, find closed formulas for the components of the dynamical system

$\stackrel{\mathbf{¯}}{\mathbf{x}}\left(t+1\right)\mathbf{=}\left[\begin{array}{cc}0.5& 0.25\\ 0.5& 0.75\end{array}\right]\stackrel{\mathbf{¯}}{\mathbf{x}}\mathbf{\left(}\mathit{t}\mathbf{\right)}$

with initial value $\stackrel{\mathbf{→}}{{\mathbf{x}}_{\mathbf{0}}}\mathbf{=}\stackrel{\mathbf{→}}{{\mathbf{e}}_{\mathbf{1}}}$. Then do the same for the initial value $\stackrel{\mathbf{→}}{{\mathbf{x}}_{\mathbf{0}}}\mathbf{=}\stackrel{\mathbf{→}}{{\mathbf{e}}_{\mathbf{2}}}$. Sketch the two trajectories.

b. Consider the matrix

$\mathit{A}\mathbf{=}\left[\begin{array}{cc}0.5& 0.25\\ 0.5& 0.75\end{array}\right]$

.

Using technology, compute some powers of the matrix A, say, A², A⁵, A¹⁰, . . . .What do you observe? Diagonalize matrix Ato prove your conjecture. (Do not use Theorem 2.3.11, which we have not proven

yet.)

c. If $\mathbf{A}\mathbf{=}\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$

is an arbitrary positive transition matrix, what can you say about the powers A^tas t goes to infinity? Your result proves Theorem 2.3.11c for the special case of a positive transition matrix of size 2 × 2.

Consider a 2 × 2 matrix A. Suppose that tr A = 5 and det A = 6. Find the eigenvalues of A.