Q23E For the linear system x鈫�(t+1)=... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 9: Q23E (page 440)

For the linear system $\overset{鈫�}{x} (t + 1) = [\begin{matrix} - 1.5 & - 1 \\ 2 & 0.5 \end{matrix}] \overset{鈫�}{x} (t)$ find the matching phase portrait.

Short Answer

Expert verified

The phase portrait corresponds to II.

Step by step solution

To find the eigenvalues

Consider the linear system as follows.

$\overset{鈫�}{x} (t + 1) = [\begin{matrix} - 1.5 & - 1 \\ 2 & 0.5 \end{matrix}] \overset{鈫�}{x} (t)$

To find the eigenvalues, evaluate $|A - 位 I| = 0$ as follows.

$|A - 位 I| = 0 [\begin{matrix} - 1.5 - 位 & - 1 \\ 2 & 0.5 - 位 \end{matrix}] = 0 (- 1.5 - 位) (0.5 - 位) - 2 (- 1) = 0 - 0.75 + 1.5 位 - 0.5 位 + 位^{2} + 2 = 0$

Simplify further as follows.

$位^{2} + 位 + 1.25 = 0 位 = \frac{- 1 卤 \sqrt{{(1)}^{2} - 4 (1) (1.25)}}{2 (1)} 位 = \frac{- 1 卤 \sqrt{1 - 5}}{2} 位 = \frac{- 1 卤 i \sqrt{2}}{2}$

Simplify further as follows.

$位_{1} = \frac{- 1 + 2 i}{2} 位_{1} = \frac{- 1}{2} + 2 i 位_{2} = \frac{- 1 - 2 i}{2} 位_{2} = \frac{- 1}{2} - 2 i$

Therefore, the eigenvalues are $位_{1} = \frac{- 1}{2} + 2 i$ and $位_{2} = \frac{- 1}{2} - 2 i$

Observation of the phase portrait

Since, the eigenvalues are a $位_{1} = \frac{- 1}{2} + 2 i$ and $位_{2} = \frac{- 1}{2} - 2 i$ also in the form as follows.

$p + i q p = - \frac{1}{2} q = 1$

This shows that, $p^{2} + q^{2} > 1$

Therefore, the phase portrait is in figure 2 as follows.

Hence, the phase portrait to the linear system $\overset{鈫�}{x} (t + 1) = [\begin{matrix} - 1.5 & - 1 \\ 2 & 0.5 \end{matrix}] \overset{鈫�}{x} (t)$ is II.

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91影视

For the linear system $\overset{鈫�}{x} (t + 1) = [\begin{matrix} - 1.5 & - 1 \\ 2 & 0.5 \end{matrix}] \overset{鈫�}{x} (t)$ find the matching phase portrait.

Short Answer

Step by step solution

To find the eigenvalues

Observation of the phase portrait

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91影视

For the linear system x鈫�(t+1)=[-1.5-120.5]x鈫�(t)find the matching phase portrait.

Short Answer

Step by step solution

To find the eigenvalues

Observation of the phase portrait

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Decision Maths

Theoretical and Mathematical Physics

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Study anywhere. Anytime. Across all devices.

For the linear system $\overset{鈫�}{x} (t + 1) = [\begin{matrix} - 1.5 & - 1 \\ 2 & 0.5 \end{matrix}] \overset{鈫�}{x} (t)$ find the matching phase portrait.