2. (3 points) Suppose that a 2×2 matrix A has an eigenvalue λ = 1 + 3i with eigenvector [2 + i 1]. Compute the general solution to y→′ = Ay→. Read more Read less
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stochastic process
![2. (3 points) Suppose that a 2×2 matrix A has an eigenvalue λ = 1 + 3i with eigenvector [2 + i 1]. Compute the general solution to y→′ = Ay→.](https://www.doubtrix.com/uploads/editor/4945043271KcicglgKfg.png)
![Suppose that the matrix A has the following eigenvalues and eigenvectors: λ1 = 1 with v→1 = [1 0]. and λ2 = −1 with v→2 = [4 1]. Write the solution to the linear system r→′ = Ar→ in the following forms. A. In eigenvalue/eigenvector form: [x(t) y(t)] = c1[ ]et + c2[ ]et B. In fundamental matrix form: [x(t)y(t)] = [ ][c1 c2] C. As two equations: (write "c1" and "c2" for c1 and c2 ) x(t) = y(t) = Note: if you are feeling adventurous you could use other eigenvectors like 4v→1 or −3v→2.](https://www.doubtrix.com/uploads/editor/2097591975WwjdwwDMAQ.png)
![Suppose that the matrix A has the following eigenvalues and eigenvectors: λ1 = 1 with v→1 = [1 0]. and λ2 = −1 with v→2 = [4 1]. Write the solution to the linear system r→′ = Ar→ in the following forms. A. In eigenvalue/eigenvector form: [x(t) y(t)] = c1[ ]et + c2[ ]et B. In fundamental matrix form: [x(t)y(t)] = [ ][c1 c2] C. As two equations: (write "c1" and "c2" for c1 and c2 ) x(t) = y(t) = Note: if you are feeling adventurous you could use other eigenvectors like 4v→1 or −3v→2.](https://www.doubtrix.com/uploads/editor/6708762412vXMQsPRlrb.png)
![5. (7 points) In this problem we'll complete the first few steps of solving a differential equation in context using eigenvectors and eigenvalues. Suppose a car rental company has two locations, location P and location Q. When a customer rents a car at one location, they have the option to return it to either location at the end of the day. Market research determines that 80% of the cars rented at location P are returned to P and 20% are returned to Q. 40% of the cars rented at location Q are returned to Q and 60% are returned to P. (a) Suppose that there are 1000 cars at location P and no cars at location Q on Monday morning. How many cars are there at location P and Q respectively at the end of the day on Monday? (b) How many cars are at each location, P and Q, at the end of the day on Tuesday? (b) How many cars are at each location, P and Q, at the end of the day on Tuesday? (c) If we let Pk and Qk be the number of cars at locations P and Q, respectively, at the end of day k, then we have Pk+1 = 0.8Pk + 0.6Qk Qk+1 = 0.2Pk + 0.4Qk Explain why this system of equations appropriately models the given situation. (d) Write the above system of equations as a matrix equation. (e) Suppose that v→ = [3 1] and w→ = [−1 1]. Compute Av→ and Aw→ to show that v→ and w→ are eigenvectors of A (where A is the coefficient matrix you found in part (d)). (f) What are the associated eigenvalues? Let λ1 be the eigenvalue associated with v→ and λ2 be the eigenvalue associated with w→. (g) We said that 1000 cars are initially at location P and none at location Q. This means that the initial vector describing the number of cars is [P0 Q0] = [1000 0] Write this initial condition as a linear combination of v→ and w→. To do this, consider [P0 Q0] = c1v→ + c2w→ and solve for c1 and c2. Then, substitute those values back into this matrix equation. [P0 Q0] = …−−−−−−[3 1]+…−−−−−−−[−1 1]](https://www.doubtrix.com/uploads/editor/6372428458LgmZXNxqka.png)
![5. (7 points) In this problem we'll complete the first few steps of solving a differential equation in context using eigenvectors and eigenvalues. Suppose a car rental company has two locations, location P and location Q. When a customer rents a car at one location, they have the option to return it to either location at the end of the day. Market research determines that 80% of the cars rented at location P are returned to P and 20% are returned to Q. 40% of the cars rented at location Q are returned to Q and 60% are returned to P. (a) Suppose that there are 1000 cars at location P and no cars at location Q on Monday morning. How many cars are there at location P and Q respectively at the end of the day on Monday? (b) How many cars are at each location, P and Q, at the end of the day on Tuesday? (b) How many cars are at each location, P and Q, at the end of the day on Tuesday? (c) If we let Pk and Qk be the number of cars at locations P and Q, respectively, at the end of day k, then we have Pk+1 = 0.8Pk + 0.6Qk Qk+1 = 0.2Pk + 0.4Qk Explain why this system of equations appropriately models the given situation. (d) Write the above system of equations as a matrix equation. (e) Suppose that v→ = [3 1] and w→ = [−1 1]. Compute Av→ and Aw→ to show that v→ and w→ are eigenvectors of A (where A is the coefficient matrix you found in part (d)). (f) What are the associated eigenvalues? Let λ1 be the eigenvalue associated with v→ and λ2 be the eigenvalue associated with w→. (g) We said that 1000 cars are initially at location P and none at location Q. This means that the initial vector describing the number of cars is [P0 Q0] = [1000 0] Write this initial condition as a linear combination of v→ and w→. To do this, consider [P0 Q0] = c1v→ + c2w→ and solve for c1 and c2. Then, substitute those values back into this matrix equation. [P0 Q0] = …−−−−−−[3 1]+…−−−−−−−[−1 1]](https://www.doubtrix.com/uploads/editor/6896791134NPlCNIwqTC.png)
![5. (7 points) In this problem we'll complete the first few steps of solving a differential equation in context using eigenvectors and eigenvalues. Suppose a car rental company has two locations, location P and location Q. When a customer rents a car at one location, they have the option to return it to either location at the end of the day. Market research determines that 80% of the cars rented at location P are returned to P and 20% are returned to Q. 40% of the cars rented at location Q are returned to Q and 60% are returned to P. (a) Suppose that there are 1000 cars at location P and no cars at location Q on Monday morning. How many cars are there at location P and Q respectively at the end of the day on Monday? (b) How many cars are at each location, P and Q, at the end of the day on Tuesday? (b) How many cars are at each location, P and Q, at the end of the day on Tuesday? (c) If we let Pk and Qk be the number of cars at locations P and Q, respectively, at the end of day k, then we have Pk+1 = 0.8Pk + 0.6Qk Qk+1 = 0.2Pk + 0.4Qk Explain why this system of equations appropriately models the given situation. (d) Write the above system of equations as a matrix equation. (e) Suppose that v→ = [3 1] and w→ = [−1 1]. Compute Av→ and Aw→ to show that v→ and w→ are eigenvectors of A (where A is the coefficient matrix you found in part (d)). (f) What are the associated eigenvalues? Let λ1 be the eigenvalue associated with v→ and λ2 be the eigenvalue associated with w→. (g) We said that 1000 cars are initially at location P and none at location Q. This means that the initial vector describing the number of cars is [P0 Q0] = [1000 0] Write this initial condition as a linear combination of v→ and w→. To do this, consider [P0 Q0] = c1v→ + c2w→ and solve for c1 and c2. Then, substitute those values back into this matrix equation. [P0 Q0] = …−−−−−−[3 1]+…−−−−−−−[−1 1]](https://www.doubtrix.com/uploads/editor/1401586756DoSHMhHJpq.png)
![5. (7 points) In this problem we'll complete the first few steps of solving a differential equation in context using eigenvectors and eigenvalues. Suppose a car rental company has two locations, location P and location Q. When a customer rents a car at one location, they have the option to return it to either location at the end of the day. Market research determines that 80% of the cars rented at location P are returned to P and 20% are returned to Q. 40% of the cars rented at location Q are returned to Q and 60% are returned to P. (a) Suppose that there are 1000 cars at location P and no cars at location Q on Monday morning. How many cars are there at location P and Q respectively at the end of the day on Monday? (b) How many cars are at each location, P and Q, at the end of the day on Tuesday? (b) How many cars are at each location, P and Q, at the end of the day on Tuesday? (c) If we let Pk and Qk be the number of cars at locations P and Q, respectively, at the end of day k, then we have Pk+1 = 0.8Pk + 0.6Qk Qk+1 = 0.2Pk + 0.4Qk Explain why this system of equations appropriately models the given situation. (d) Write the above system of equations as a matrix equation. (e) Suppose that v→ = [3 1] and w→ = [−1 1]. Compute Av→ and Aw→ to show that v→ and w→ are eigenvectors of A (where A is the coefficient matrix you found in part (d)). (f) What are the associated eigenvalues? Let λ1 be the eigenvalue associated with v→ and λ2 be the eigenvalue associated with w→. (g) We said that 1000 cars are initially at location P and none at location Q. This means that the initial vector describing the number of cars is [P0 Q0] = [1000 0] Write this initial condition as a linear combination of v→ and w→. To do this, consider [P0 Q0] = c1v→ + c2w→ and solve for c1 and c2. Then, substitute those values back into this matrix equation. [P0 Q0] = …−−−−−−[3 1]+…−−−−−−−[−1 1]](https://www.doubtrix.com/uploads/editor/9702922078OiTwyPXqkE.png)
![2.28. Consider the weather model of Conceptual Problem 2.13. Compute the longrun fraction of days that are sunny. 2.13. Consider the following weather model. The weather normally behaves as in Example 2.3. However, when the cloudy spell lasts for two or more days, it continues to be cloudy for another day with probability .8 or turns rainy with probability .2. Develop a four-state DTMC model to describe this behavior. Example 2.3. (Weather Model). The weather in the city of Heavenly is classified as sunny, cloudy, or rainy. Suppose that tomorrow’s weather depends only on today’s weather as follows: if it is sunny today, it is cloudy tomorrow with probability .3 and 2.2 Examples of Markov Models 9 rainy with probability . 2 ; if it is cloudy today, it is sunny tomorrow with probability .5 and rainy with probability .3 ; and finally, if it is rainy today, it is sunny tomorrow with probability .4 and cloudy with probability . 5 . Model the weather process as a DTMC. Let Xn be the weather conditions in Heavenly on day n, defined as follows: Xn = { 1 if it is sunny on day n, 2 if it is cloudy on day n, 3 if it is rainy on day n. Then we are told that {Xn, n ≥ 0} is a DTMC with state space {1, 2, 3}. We next compute its transition matrix. We are given that p1,2 = .3 and p1,3 = .2. We are not explicitly given p1,1. We use p1,1 + p1,2 + p1,3 = 1 to obtain p1,1 = .5. Similarly, we can obtain p2,2 and p3,3. This yields the transition probability matrix P = [.50 .30 .20 .50 .20 .30 .40 .50 .10]](https://www.doubtrix.com/uploads/editor/8113895426FgPKVyKTLr.png)
![2.28. Consider the weather model of Conceptual Problem 2.13. Compute the longrun fraction of days that are sunny. 2.13. Consider the following weather model. The weather normally behaves as in Example 2.3. However, when the cloudy spell lasts for two or more days, it continues to be cloudy for another day with probability .8 or turns rainy with probability .2. Develop a four-state DTMC model to describe this behavior. Example 2.3. (Weather Model). The weather in the city of Heavenly is classified as sunny, cloudy, or rainy. Suppose that tomorrow’s weather depends only on today’s weather as follows: if it is sunny today, it is cloudy tomorrow with probability .3 and 2.2 Examples of Markov Models 9 rainy with probability . 2 ; if it is cloudy today, it is sunny tomorrow with probability .5 and rainy with probability .3 ; and finally, if it is rainy today, it is sunny tomorrow with probability .4 and cloudy with probability . 5 . Model the weather process as a DTMC. Let Xn be the weather conditions in Heavenly on day n, defined as follows: Xn = { 1 if it is sunny on day n, 2 if it is cloudy on day n, 3 if it is rainy on day n. Then we are told that {Xn, n ≥ 0} is a DTMC with state space {1, 2, 3}. We next compute its transition matrix. We are given that p1,2 = .3 and p1,3 = .2. We are not explicitly given p1,1. We use p1,1 + p1,2 + p1,3 = 1 to obtain p1,1 = .5. Similarly, we can obtain p2,2 and p3,3. This yields the transition probability matrix P = [.50 .30 .20 .50 .20 .30 .40 .50 .10]](https://www.doubtrix.com/uploads/editor/6681069121vdqyOAMGrx.png)
![2.28. Consider the weather model of Conceptual Problem 2.13. Compute the longrun fraction of days that are sunny. 2.13. Consider the following weather model. The weather normally behaves as in Example 2.3. However, when the cloudy spell lasts for two or more days, it continues to be cloudy for another day with probability .8 or turns rainy with probability .2. Develop a four-state DTMC model to describe this behavior. Example 2.3. (Weather Model). The weather in the city of Heavenly is classified as sunny, cloudy, or rainy. Suppose that tomorrow’s weather depends only on today’s weather as follows: if it is sunny today, it is cloudy tomorrow with probability .3 and 2.2 Examples of Markov Models 9 rainy with probability . 2 ; if it is cloudy today, it is sunny tomorrow with probability .5 and rainy with probability .3 ; and finally, if it is rainy today, it is sunny tomorrow with probability .4 and cloudy with probability . 5 . Model the weather process as a DTMC. Let Xn be the weather conditions in Heavenly on day n, defined as follows: Xn = { 1 if it is sunny on day n, 2 if it is cloudy on day n, 3 if it is rainy on day n. Then we are told that {Xn, n ≥ 0} is a DTMC with state space {1, 2, 3}. We next compute its transition matrix. We are given that p1,2 = .3 and p1,3 = .2. We are not explicitly given p1,1. We use p1,1 + p1,2 + p1,3 = 1 to obtain p1,1 = .5. Similarly, we can obtain p2,2 and p3,3. This yields the transition probability matrix P = [.50 .30 .20 .50 .20 .30 .40 .50 .10]](https://www.doubtrix.com/uploads/editor/4508513795wgVUEBlqnS.png)