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Lecture Notes for ECN 200E - Macro Theory with Salyer at UC Davis (UCD)

Notes Information

Material Type:Class Note
Professor:Salyer
Class:ECN 200E - Macro Theory
Subject:Economics
University:University of California - Davis
Term:Spring 2003
Keywords:
  • Eigenvalues
  • Probabilities
  • Combination
  • Markov Process
  • Conditional
  • Interpreting
  • Eigenvector
  • State Vector
  • Eigenvectors
  • Left-Hand Side
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Interpreting the Eigenvalues in a Symmetric Stochastic Matrix Kevin D. Salyer April 11, 2003 Consider the following n-state Markov process for the random variable, x t : x t = ? ? ? ? ? ? ? x 1 x 2 . . . x n (1) The one-period transition probability matrix, with the entry in the ith row and jth column denoting the conditional probability of going from state i to state j,is: ? = ? ? ? ? ? ? ? ? 1?? n?1 1?? n?1 иии 1?? n?1 1?? n?1 ? иии иии 1?? n?1 1?? n?1 иии . . . ... 1?? n?1 1?? n?1 иии иии ? 1?? n?1 1?? n?1 иии иии иии ? ? ? ? ? ? ? ? (2) There will be n eigenvalues associated with the stochastic matrix but since the columns (and rows) sum to one, we know that one of the eigenvalues will be equal to 1. (The proof is easy: ?и1 = 1). The unconditional probabilities, p 0 =(p 1, p 2 ,...,p n ) aregivenbythesolutionto? T p = p.Sinceamatrixand its transpose have the same eigenvalues, we see that the eigenvector associated w...

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