91影视

TRUE OR FALSE

56. If$\stackrel{\mathbf{鈫�}}{\mathbf{u}}$is a nonzero vector in${\mathbf{鈩�}}^{\mathbf{n}}$, then$\stackrel{\mathbf{鈫�}}{\mathbf{u}}$ must be an eigenvector of matrix$\stackrel{\mathbf{鈫�}}{\mathbf{u}}{\stackrel{\mathbf{鈫�}}{\mathbf{u}}}^{\mathbf{T}}$.

TRUE OR FALSE

58. Ifis an eigenvector of a 2x2matrix$\mathit{A}\mathbf{=}\mathbf{\left[}\begin{array}{cc}\mathbf{a}& \mathbf{b}\\ \mathbf{c}& \mathbf{d}\end{array}\mathbf{\right]}$, then$\stackrel{\mathbf{鈫�}}{\mathbf{v}}$must be an eigenvector of its classical adjoint$\mathit{A}\mathbf{=}\mathbf{\left[}\begin{array}{cc}\mathbf{d}& \mathbf{-}\mathbf{b}\\ \mathbf{-}\mathbf{c}& \mathbf{a}\end{array}\mathbf{\right]}$as well.

Consider an invertible n 脳 n matrix A such that the zero state is a stable equilibrium of the dynamical system$\stackrel{\mathbf{鈫�}}{\mathbf{x}}\mathbf{(}\mathit{t}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{=}\mathit{A}\stackrel{\mathbf{鈫�}}{\mathbf{x}}\mathbf{\left(}\mathit{t}\mathbf{\right)}$.What can you say about the stability of the systems$\stackrel{\mathbf{鈫�}}{\mathbf{x}}\mathbf{(}\mathit{t}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{=}\left(A+I_n\right)\stackrel{\mathbf{鈫�}}{\mathbf{x}}\mathbf{\left(}\mathit{t}\mathbf{\right)}$

Consider an invertible n 脳 n matrix A such that the zero state is a stable equilibrium of the dynamical system$\stackrel{\mathbf{鈫�}}{\mathbf{x}}\mathbf{(}\mathit{t}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{=}\mathit{A}\stackrel{\mathbf{鈫�}}{\mathbf{x}}\mathbf{\left(}\mathit{t}\mathbf{\right)}$.What can you say about the stability of the systems

$\stackrel{\mathbf{鈫�}}{\mathbf{x}}\mathbf{(}\mathit{t}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{=}{\mathbf{A}}^{\mathbf{2}}\stackrel{\mathbf{鈫�}}{\mathbf{x}}\mathbf{\left(}\mathit{t}\mathbf{\right)}$

Let A be a real 2 脳 2 matrix. Show that the zero state is a stable equilibrium of the dynamical system$\stackrel{\mathbf{鈫�}}{\mathbf{x}}\mathbf{(}\mathit{t}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{=}\mathit{A}\stackrel{\mathbf{鈫�}}{\mathbf{x}}\mathbf{\left(}\mathit{t}\mathbf{\right)}$if (and only if)$\mathbf{\left|}\mathit{t}\mathit{r}\mathit{A}\mathbf{\right|}\mathbf{-}\mathbf{1}\mathbf{<}\mathit{d}\mathit{e}\mathit{t}\mathit{A}\mathbf{<}\mathbf{1}$

Let鈥檚 revisit the introductory example of Section 7.5: The glucose regulatory system of a certain patient can be modeled by the equations

$$\begin{gathered} \mathit{g}\mathbf{(}\mathit{t}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{9}\mathit{g}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{4}\mathit{h}\mathbf{\left(}\mathit{t}\mathbf{\right)} \\ \mathit{h}\mathbf{(}\mathit{t}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{1}\mathit{g}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{9}\mathit{h}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{.} \end{gathered}$$

Find closed formulas for g(t) and h(t), and draw the trajectory. Does your trajectory look like the one on page 361?

Consider a real 2 脳 2 matrix A with eigenvalues $\mathit{p}\mathbf{卤}\mathit{i}\mathit{q}$and corresponding eigenvectors$\stackrel{\mathbf{鈫�}}{\mathbf{v}}\mathbf{卤}\mathit{i}\stackrel{\mathbf{鈫�}}{\mathbf{w}}$.Show that if a real vector${\stackrel{\mathbf{鈫�}}{\mathbf{x}}}_{\mathbf{0}}$is written as${\stackrel{\mathbf{鈫�}}{\mathbf{x}}}_{\mathbf{0}}\mathbf{=}{\mathit{c}}_{\mathbf{1}}\mathbf{(}\stackrel{\mathbf{鈫�}}{\mathbf{v}}\mathbf{+}\mathit{i}\stackrel{\mathbf{鈫�}}{\mathbf{w}}\mathbf{)}\mathbf{+}{\mathit{c}}_{\mathbf{2}}\mathbf{(}\stackrel{\mathbf{鈫�}}{\mathbf{v}}\mathbf{-}\mathit{i}\stackrel{\mathbf{鈫�}}{\mathbf{w}}\mathbf{)}$then ${\mathit{c}}_{\mathbf{2}}\mathbf{=}{\stackrel{\mathbf{炉}}{\mathbf{c}}}_{\mathbf{1}}$.

(a). Consider a real n 脳 n matrix with n distinct real eigenvalues${\mathit{位}}_{\mathbf{1}}\mathbf{,}\mathbf{鈥�}\mathbf{,}{\mathit{位}}_{\mathbf{n}}$where$\left|{位}_i\right|\mathbf{猢�}\mathbf{1}$for all i =$\mathbf{1}\mathbf{,}\mathbf{鈥�}\mathbf{,}\mathit{n}$. Letbe a trajectory of the dynamical system$\stackrel{\mathbf{鈫�}}{\mathbf{x}}\mathbf{(}\mathit{t}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{=}\mathit{A}\stackrel{\mathbf{鈫�}}{\mathbf{x}}\mathbf{\left(}\mathit{t}\mathbf{\right)}$. Show that this trajectory is bounded; that is, there is a positive number M such that$\mathbf{鈭�}\stackrel{\mathbf{鈫�}}{\mathbf{x}}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{鈭�}\mathbf{猢�}\mathit{M}$for all positive integers t.

(b). Are all trajectories of the dynamical system

$\stackrel{\mathbf{鈫�}}{\mathbf{x}}\mathbf{(}\mathit{t}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{=}\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)\stackrel{\mathbf{鈫�}}{\mathbf{x}}\mathbf{\left(}\mathit{t}\mathbf{\right)}$

bounded? Explain.

Show that the zero state is a stable equilibrium of the dynamical system $\stackrel{\mathbf{鈫�}}{\mathbf{x}}\mathbf{(}\mathit{t}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{=}\mathit{A}\stackrel{\mathbf{鈫�}}{\mathbf{x}}\mathbf{\left(}\mathit{t}\mathbf{\right)}$if (and only if )$\underset{\mathbf{t}\mathbf{鈫�}\mathbf{鈭�}}{\mathbf{l}\mathbf{i}\mathbf{m}}{\mathit{A}}^{\mathbf{t}}\mathbf{=}\mathbf{0}$(meaning that all entries of${\mathit{A}}^{\mathbf{t}}$approach zero).

Consider the national income of a country, which consists of consumption, investment, and government expenditures. Here we assume the government expenditure to be constant, at G₀, while the national income Y(t), consumption C(t) , and investment I(t) change over time. According to a simple model, we have

$$\begin{gathered} |Y\left(t\right)=C\left(t\right)+I\left(t\right)+G_0 \\ C\left(t+1\right)=纬Y\left(t\right) \\ I\left(t+1\right)=伪\left(C\left(t+1\right)-C\left(t\right)\right)| \end{gathered}$$ $$\begin{gathered} \left(0<纬<1\right)\mathbf{,} \\ \left(伪>0\right) \end{gathered}$$

Where $\mathit{纬}$is the marginal propensity to consume and $\mathit{伪}$is the acceleration coefficient. (See Paul E. Samuelson, 鈥淚nteractions between the Multiplier Analysis and the Principle of Acceleration,鈥� Review of Economic Statistics, May 1939, pp. 75-78.)

1. Find the equilibrium solution of these equations, when $\mathit{Y}\left(t+1\right)\mathbf{=}\mathit{Y}\left(t,\right)\mathbf{}\mathit{C}\left(t+1\right)\mathbf{=}\mathit{C}\left(t\right)\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\mathit{I}\left(t+1\right)\mathbf{=}\mathit{I}\left(t\right)$.
2. Let $\mathit{y}\left(t\right)\mathbf{,}\mathbf{}\mathit{c}\left(t\right)\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\mathit{i}\left(t\right)$be the deviations of$\mathit{Y}\left(t\right)\mathbf{,}\mathbf{}\mathit{C}\left(t\right)\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\mathit{I}\left(t\right)$, , and , respectively, from the equilibrium state you found in part (a). These quantities are related by the equations
   $$\begin{gathered} \left|y\left(t\right)=c\left(t\right)+i\left(t\right) \\ c\left(t+1\right)=纬y\left(t\right) \\ i\left(t+1\right)=伪\left(c\left(t+1\right)-c\left(t\right)\right)\right| \end{gathered}$$
   (Verify this!) By substituting y(t) into the second equations, set up equations of the form
   $$\begin{gathered} \left|c\left(t+1\right)=pc\left(t\right)+qi\left(t\right) \\ i\left(t+1\right)=rc\left(t\right)+si\left(t\right)\right| \end{gathered}$$
   
3. When $\mathit{伪}\mathbf{=}\mathbf{5}$and $\mathit{纬}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{2}$, determine the stability of zero state of this system.
4. When $\mathit{伪}\mathbf{=}\mathbf{1}$(and $\mathit{纬}$is arbitrary, $\mathbf{0}\mathbf{<}\mathit{纬}\mathbf{<}\mathbf{1}$), determine the stability of the zero state.
5. For each of the four sectors in the $\mathit{伪}\mathbf{-}\mathit{纬}\mathbf{-}$plane, determine the stability of the zero state.

Discuss the various cases, in practical terms.