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Linear Algebra With Applications
Found in: Page 336
Linear Algebra With Applications

Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

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Short Answer

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

x¯(t+1)=[0.5 0.250.5 0.75]x¯(t)

with initial value x0=e1. Then do the same for the initial value x0=e2. Sketch the two trajectories.

b. Consider the matrix

A=[0.5 0.250.5 0.75]


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


c. If A=[abcd]

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

  1. Formulas for the dynamical system :ct=13.1t+230.25t=13+230.25trt=23.1t-230.25t=13-230.25t

  1. An=13132323
  2. llimtAt=1b+cbbcc
See the step by step solution

Step by Step Solution

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.

Step 2: Finding closed formulas for the dynamical system and computing some powers of the matrix :

(a) Clearly we see that,


(b) We compute,


(c) We can see that for t>0 applies that,


Hence, the final answer is : (a) ct=13.1t+230.25t=13+230.25trt=23.1t-230.25t=13-230.25t

(b) An=13132323

(c llimtAt=1b+cbbcc

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