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3. The city of Mawroke maintains a constant voter of 300,000 people per year. Apolitical science study estimated that there were 150,000 Independents, 90,000 Democrats, and 60,000 Republicans in the city. It was also estimated that each year 20% of the Independents become Democrats and 10% become Republicans. Similarly, 20% of Democrats become Independents and 10% become Republicans, while 10% of Republicans defect to the Democrats and 10% become Independents each year. (a) Find stochastic transition matrix A such that æ* = Ark-¹, k = 1,2,3.., where ä³ be a vector representing the number of people in each group. (b) Find the eigenvalues of A and corresponding eigenvectors vectors. (c) Factor A into a PDP-1, where D is diagonal. (d) Which group will dominate in the long run? Justify your answer by computing lim A". nx

3. The city of Mawroke maintains a constant voter of 300,000 people per year. Apolitical science study estimated that there were 150,000 Independents, 90,000 Democrats, and 60,000 Republicans in the city. It was also estimated that each year 20% of the Independents become Democrats and 10% become Republicans. Similarly, 20% of Democrats become Independents and 10% become Republicans, while 10% of Republicans defect to the Democrats and 10% become Independents each year. (a) Find stochastic transition matrix A such that æ* = Ark-¹, k = 1,2,3.., where ä³ be a vector representing the number of people in each group. (b) Find the eigenvalues of A and corresponding eigenvectors vectors. (c) Factor A into a PDP-1, where D is diagonal. (d) Which group will dominate in the long run? Justify your answer by computing lim A". nx

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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3. The city of Mawroke maintains a constant voter population of 300,000 people per year. A political science study estimated that there were 150,000 Independents, 90,000 Democrats, and 60,000 Republicans in the city. It was also estimated that each year 20% of the Independents become Democrats and 10% become Republicans. Similarly, 20% of Democrats become Independents and 10% become Republicans, while 10% of Republicans defect to the Democrats and 10% become Independents each year. (a) Find the stochastic transition matrix \( A \) such that \( x^k = A x^{k-1} \), \( k = 1, 2, 3, \ldots \), where \( x^k \) is a vector representing the number of people in each group. (b) Find the eigenvalues of \( A \) and corresponding eigenvectors. (c) Factor \( A \) into a \( PDP^{-1} \), where \( D \) is diagonal. (d) Which group will dominate in the long run? Justify your answer by computing \( \lim_{n \to \infty} A^n x \). (Note: Above all, a blue stain or mark is present on the right side of the text.)
Transcribed Image Text:3. The city of Mawroke maintains a constant voter population of 300,000 people per year. A political science study estimated that there were 150,000 Independents, 90,000 Democrats, and 60,000 Republicans in the city. It was also estimated that each year 20% of the Independents become Democrats and 10% become Republicans. Similarly, 20% of Democrats become Independents and 10% become Republicans, while 10% of Republicans defect to the Democrats and 10% become Independents each year. (a) Find the stochastic transition matrix \( A \) such that \( x^k = A x^{k-1} \), \( k = 1, 2, 3, \ldots \), where \( x^k \) is a vector representing the number of people in each group. (b) Find the eigenvalues of \( A \) and corresponding eigenvectors. (c) Factor \( A \) into a \( PDP^{-1} \), where \( D \) is diagonal. (d) Which group will dominate in the long run? Justify your answer by computing \( \lim_{n \to \infty} A^n x \). (Note: Above all, a blue stain or mark is present on the right side of the text.)
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