Designing Information Devices and Systems I Spring 2019 Homework 5

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1 Last Updated: :38 EECS 6A Designing Information Devices Systems I Spring 209 Homework 5 This homework is due March, 209, at 23:59. Self-grades are due March 5, 209, at 23:59. Submission Format Your homework submission should consist of one file. hw5.pdf: A single PDF file that contains all of your answers any hwritten answers should be scanned) Submit the file to the appropriate assignment on Gradescope.. The Dynamics of Romeo Juliet s Love Affair In this problem, we will study a discrete-time model of the dynamics of Romeo Juliet s love affair adapted from Steven H. Strogatz s original paper, Love Affairs Differential Equations, Mathematics Magazine, 6), p.35, 988, which describes a continuous-time model. Let R[n] denote Romeo s feelings about Juliet on day n, let J[n] denote Juliet s feelings about Romeo on day n. The sign of R[n] or J[n]) indicates like or dislike. For example, if R[n] > 0, it means Romeo likes Juliet. On the other h, R[n] < 0 indicates that Romeo dislikes Juliet. R[n] = 0 indicates that Romeo has a neutral stance towards Juliet. The magnitude i.e. absolute value) of R[n] or J[n]) represents the intensity of that feeling. For example, a larger R[n] means that Romeo has a stronger emotion towards Juliet love if R[n] > 0 or hatred if R[n] < 0). Similar interpretations hold for J[n]. We model the dynamics of Romeo Juliet s relationship using the following linear system: which we can rewrite as R[n + ] = ar[n] + bj[n], n = 0,,2,... J[n + ] = cr[n] + d J[n], n = 0,,2,..., s[n + ] = A s[n], [ ] [ ] R[n] a b where s[n] = denotes the state vector the state transition matrix for our dynamic J[n] c d system model. The selection of the parameters a, b, c, d results in different dynamic scenarios. The fate of Romeo Juliet s relationship depends on these model parameters i.e. a, b, c, d) in the state transition matrix the initial state s[0]). In this problem, we ll explore some of these possibilities. UCB EECS 6A, Spring 209, Homework 5, All Rights Reserved. This may not be publicly shared without explicit permission.

2 Last Updated: :38 2 a) Consider the case where a + b = c + d in the state-transition matrix [ ] a b. c d Show that v = is an eigenvector of A, determine its corresponding eigenvalue λ. Show that v 2 = is an eigenvector of A, determine its corresponding eigenvalue λ 2. Now, express the first second eigenvalues their eigenspaces in terms of the parameters a,b,c, d. Hint: You could use the characteristic polynomial approach to find the eigenvalues eigenvectors. You may find it easier to use the following approach instead: First find λ by showing v = is an eigenvector of A. Then find λ 2 by showing v 2 = is an eigenvector of A. [ ] [ ] a + b c + d = a + b) = c + d) Let λ = a+b = c+d. Then you can plug in to find that [ to the eigenvalue λ. The first eigenpair A is, λ = a + b = c + d, v = ] T is an eigenvector of A corresponding [ ]) To determine the other eigenpair λ 2, v 2 ), we use the hint that v 2 =. Note that by modifying the constraint a + b = c + d, we can also get a c = d b, which helps simplify the following: [ ] [ ] b ab bc cb dc a c) = d b) = a c) = d b) UCB EECS 6A, Spring 209, Homework 5, All Rights Reserved. This may not be publicly shared without explicit permission. 2

3 Last Updated: :38 3 Therefore, we have our second eigenpair: [ ]) b λ 2 = a c = d b, v 2 =. For parts b) - d), consider the following state-transition matrix: [ ] b) Determine the eigenpairs i.e. λ, v ) λ 2, v 2 )) for this system. Note that this matrix is a special case of the matrix explored in part a), so you can use results from that part to help you. From the results of part a), we know that the eigenpairs of this matrix are λ = a + b = =, v = [ ]) [ ]) λ 2 = a c = = 0.5, v 2 =. Note: If your choice of eigenvector v v 2 is a scaled version of the ones given in this solution, that is fine. c) Determine all of the steady states of the system. That is, find the set of points such that if Romeo Juliet start at, or enter, any of those points, their states will stay in place forever: { s A s = s }. Any s span{ v }, where v =, is the eigenvector which corresponds to the steady state, because v corresponds to the eigenvalue λ =. {[ ]} d) Suppose Romeo Juliet start from an initial state s[0] span. What happens to their relationship over time? Specifically, what is s[n] as n? We note that s[0] span{ v 2 }. Therefore, s[] = A s[0] = αλ 2 v 2 where α is the scalar that expresses s[0] as a scaled version of v 2. If we continue to apply the state transition matrix, we will see that for this s[0], = αλ n 2 v 2 In this case λ 2 = 0.5. This means that as n, λ n 2 0. UCB EECS 6A, Spring 209, Homework 5, All Rights Reserved. This may not be publicly shared without explicit permission. 3

4 Last Updated: :38 4 Therefore, which means that s[n] = αλ n 2 v 2 = α 0 v 2 = 0 lim R[n],J[n]) = 0,0) n So, ultimately, Romeo Juliet will become neutral to each other. Now suppose we have the following state-transition matrix: Use this state-transition matrix for parts e) - g). e) Determine the eigenpairs i.e. λ, v ) λ 2, v 2 )) for this system. Note that this matrix is a special case of the matrix explored in part a), so you can use results from that part to help you. From the results of part a), we know that the eigenpairs of this matrix are λ = a + b = + = 2, v = [ ]) [ ]) λ 2 = a c = = 0, v 2 =. {[ ]} f) Suppose Romeo Juliet start from an initial state s[0] span. What happens to their relationship over time? Specifically, what is s[n] as n? The initial state s[0] lies in the span of the eigenvector v 2, which has eigenvalue λ 2 = 0. Thus, s[] = 0. The state will remain at 0 for all subsequent time steps, i.e. s[n] = 0,n Therefore, Romeo Juliet become neutral towards each other in the long run, i.e. lim R[n],J[n]) = 0,0) n g) Now suppose that Romeo Juliet start from an initial state s[0] span their relationship over time? Specifically, what is s[n] as n? {[ ]}. What happens to UCB EECS 6A, Spring 209, Homework 5, All Rights Reserved. This may not be publicly shared without explicit permission. 4

5 Last Updated: :38 5 We note that s[0] span{ v }. Therefore, s[] = A s[0] = αλ v where α is the scalar that expresses s[0] as a scaled version of v. If we continue to apply the state transition matrix, we will see that for this s[0], In this problem, λ = 2. Therefore, = αλ n v s[n] = α2 n v This means that as n, λ n. Essentially, the elements of the state vector continue to double at each time step grow without bound to either + or. Therefore, what happens to Romeo Juliet depends on s[0]. If s[0] is in the first quadrant, Romeo Juliet will become infinitely in love with each other. On the other h, if s[0] is in the third quadrant, then Romeo Juliet will have infinite hatred for each other. Finally, we consider the case where we have the following state-transition matrix: 2 2 Use this state-transition matrix for parts h) - j). h) Determine the eigenpairs i.e. λ, v ) λ 2, v 2 )) for this system. Note that this matrix is a special case of the matrix explored in part a), so you can use results from that part to help you. From the results of part a), we know that the eigenpairs of this matrix are λ = a + b = 2 =, v = [ ]) [ ]) λ 2 = a c = 2) = 3, v 2 =. {[ ]} i) Suppose Romeo Juliet start from an initial state s[0] span. What happens to their relationship over time if R[0] > 0 J[0] < 0? What about if R[0] < 0 J[0] > 0? Specifically, what is s[n] as n? The initial state s[0] lies in the span of the eigenvector v 2, which has eigenvalue λ 2 = 3. Using similar methods to the solutions in part d) part g), we can see that for a given scalar α): UCB EECS 6A, Spring 209, Homework 5, All Rights Reserved. This may not be publicly shared without explicit permission. 5

6 Last Updated: :38 6 = αλ n 2 v 2 = α3 n v 2 There are two cases of long-term behavior. Suppose, initially, that R[0] > 0 J[0] < 0 corresponding to α > 0). Then as n, R[n] J[n]. Romeo will have infinite love for Juliet, while Juliet will have infinite hatred for Romeo. Conversely, if initially R[0] < 0 J[0] > 0 corresponding to α < 0), then as n, R[n] J[n]. Now Romeo would have infinite hatred for Juliet, while Juliet would have infinite love for Romeo. j) Now suppose that Romeo Juliet start from an initial state s[0] span {[ ]}. What happens to their relationship over time? Specifically, what is s[n] as n? The initial state s[0] lies in the span of the eigenvector v, which has eigenvalue λ =. As with parts d), g), i), we can see that for a given scalar α): = αλ n v = α ) n v The elements of the state vector continue to switch signs at each time step, while keeping the same magnitude. Essentially, Romeo Juliet maintain the same intensity i.e. absolute value or magnitude) of feeling, but they keep changing their mind about whether that feeling is like or dislike at each time step. Note that R[0] J[0] have the same sign, so they both either like each other or dislike each other at a given time step n. 2. Homework Process Study Group Who else did you work with on this homework? List names student ID s. In case of homework party, you can also just describe the group.) How did you work on this homework? I worked on this homework with... I first worked by myself for 2 hours, but got stuck on problem 5, so I went to office hours on... Then I went to homework party for a few hours, where I finished the homework. 3. Midterm Problem 3 Redo Midterm Problem 3. a) See midterm solutions. b) See midterm solutions. c) See midterm solutions. UCB EECS 6A, Spring 209, Homework 5, All Rights Reserved. This may not be publicly shared without explicit permission. 6

7 Last Updated: : Midterm Problem 4 Redo Midterm Problem 4. a) See midterm solutions. b) See midterm solutions. 5. Midterm Problem 5 Redo Midterm Problem 5. a) See midterm solutions. b) See midterm solutions. c) See midterm solutions. d) See midterm solutions. 6. Midterm Problem 6 Redo Midterm Problem 6. a) See midterm solutions. b) See midterm solutions. c) See midterm solutions. d) See midterm solutions. 7. Midterm Problem 7 Redo Midterm Problem 7. a) See midterm solutions. b) See midterm solutions. c) See midterm solutions. 8. Midterm Problem 8 Redo Midterm Problem 8. a) See midterm solutions. b) See midterm solutions. c) See midterm solutions. UCB EECS 6A, Spring 209, Homework 5, All Rights Reserved. This may not be publicly shared without explicit permission. 7

Designing Information Devices and Systems I Fall 2018 Homework 5

Last Updated: 08-09-9 0:6 EECS 6A Designing Information Devices and Systems I Fall 08 Homework 5 This homework is due September 8, 08, at 3:59. Self-grades are due October, 08, at 3:59. Submission Format