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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....
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 Spring '08
 ee221a
 Linear Algebra, Matrices, Variance, Eigenvalue, eigenvector and eigenspace, Stationary point

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