Short Answer

26: Based on your answers in Exercises 24 and 25, sketch a phase portrait of the dynamical system
_{$\bar{x} (t + 1) = [\begin{matrix} 0.5 & 0.25 \\ 0.5 & 0.75 \end{matrix}] \bar{x} (t)$}

Phase portrait of a dynamical system is: $A^{t} x_{0} = [\begin{matrix} 0 . 25^{t} α + 0.25 β \\ - 0 . 25^{t} α + 0.5 β \end{matrix}]$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Transition Matrix

Transition matrix may refer to: The matrix associated with a change of basis for a vector space. Stochastic matrix, a square matrix used to describe the transitions of a Markov chain. State-transition matrix, a matrix whose product with the state vector at an initial time gives state vector at that time.

Step 2: Sketching a phase portrait of the dynamical system:

As we clearly now that,

$v_{1} = [\begin{matrix} 1 \\ - 1 \end{matrix}] v_{2} = [\begin{matrix} 0.25 \\ 0.5 \end{matrix}]$

role="math" localid="1659584611493" $x_{0} = α [\begin{matrix} 1 \\ - 1 \end{matrix}] + β [\begin{matrix} 0.25 \\ 0.5 \end{matrix}] A^{t} x_{0} = 0 . 25^{t} α [\begin{matrix} 1 \\ - 1 \end{matrix}] + β [\begin{matrix} 0.25 \\ 0.5 \end{matrix}] = [\begin{matrix} 0 . 25^{t} α + 0.25 β \\ - 0 . 25^{t} α + 0.5 β \end{matrix}]$

Hence, the final answer is: $A^{t} x_{0} = [\begin{matrix} 0 . 25^{t} α + 0.25 β \\ - 0 . 25^{t} α + 0.5 β \end{matrix}]$

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Linear Algebra With Applications

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Short Answer

26: Based on your answers in Exercises 24 and 25, sketch a phase portrait of the dynamical system
_{$\bar{x} (t + 1) = [\begin{matrix} 0.5 & 0.25 \\ 0.5 & 0.75 \end{matrix}] \bar{x} (t)$}

Step by Step Solution

Step 1: Transition Matrix

Step 2: Sketching a phase portrait of the dynamical system:

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Linear Algebra With Applications

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26: Based on your answers in Exercises 24 and 25, sketch a phase portrait of the dynamical systemx¯(t+1)=[0.50.250.50.75]x¯(t)

Step by Step Solution

Step 1: Transition Matrix

Step 2: Sketching a phase portrait of the dynamical system:

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26: Based on your answers in Exercises 24 and 25, sketch a phase portrait of the dynamical system
_{$\bar{x} (t + 1) = [\begin{matrix} 0.5 & 0.25 \\ 0.5 & 0.75 \end{matrix}] \bar{x} (t)$}