day3-soln

day3-soln - Review Problems 3 iCME and MS&E Refresher...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Review Problems 3 iCME and MS&E Refresher Course Wednesday, 15 September, 2010 1. Markov Matrices: Suppose that each year 10% of the people outside California move in and 20% of the people inside California move out. We start with y people outside and z people inside. At the end of the first year the numbers outside and inside are y 1 and z 1 : y 1 = . 9 y + 0 . 2 z z 1 = . 1 y + 0 . 8 z or alternatively [ y 1 z 1 ] = [ . 9 . 2 . 1 . 8 ][ y z ] This problem and its matrix have two essential properties of a Markov process. (a) The total number of people stay fixed: Each column of the Markov matrix adds up to 1. Nobody is gained or lost. (b) The numbers in this system can never become negative: The matrix has no negative entries. Also, the powers A k are all nonnegative. Answer the following questions: (a) What are the eigenvalues λ i and eigenvectors x i of the matrix A ? Verify the eigenvalue decomposition for the matrix A , i.e. show that we can write A = S Λ S- 1 where Λ = [ λ 1 λ 2 ] and the columns of S are the eigenvectors x i . Solution : It is easy to show that λ 1 = 1 , λ 2 = 0 . 7 x 1 = 2 3 1 3 x 2 = 1 3 − 1 3 1 So that we have the eigenvalue decomposition A = S Λ S- 1 = 2 3 1 3 1 3 − 1 3 1 . 7 1 1 1 − 2 (b) Show that (when the eigenvalue decomposition exists) A k = A × A × ··· × A = S Λ k S- 1 Solution : A k = ( S Λ S- 1 )( S Λ S- 1 ) ... ( S Λ S- 1 ) = S Λ × Λ ×···× Λ S- 1 = S Λ k S- 1 (c) Give an explicit formula for the powers A k when A = [ . 9 . 2 . 1 . 8 ] Solution : A = S Λ k S- 1 = 2 3 1 3 1 3 − 1 3 1 k . 7 k 1 1 1 − 2 (d) Define [ y k z k ] = A [ y k- 1 z k- 1 ] that is, the population every year only depends on the previous year. What is [ y k z k ] in terms of [ y z ] ? Solution : y k z k = A k [ y z ] (e) Now let k → ∞ . What is [ y ∞ z ∞ ] ? This is known as the steady state of the system. Solution : As k → ∞ , 1 k → 1 and 0 . 7 k → 0 so that, after performing the multiplication, [ y k z k ] = ( y + z ) [ 2 3 1 3 ] (f) Verify that A [ y ∞ z ∞ ] = [ y ∞ z ∞ ] in other words, the steady state is the eigenvector corresponding to eigenvalue 1....
View Full Document

This note was uploaded on 10/01/2011 for the course EE 221 taught by Professor Ee221a during the Spring '08 term at Berkeley.

Page1 / 8

day3-soln - Review Problems 3 iCME and MS&E Refresher...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online