# 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: EigenvaluesProbabilitiesCombinationMarkov ProcessConditionalInterpretingEigenvectorState VectorEigenvectorsLeft-Hand Side  ## Sample Document Text

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 ��� 1?? n?1 1?? n?1 ? ��� ��� 1?? n?1 1?? n?1 ��� . . . ... 1?? n?1 1?? n?1 ��� ��� ? 1?? n?1 1?? n?1 ��� ��� ��� ? ? ? ? ? ? ? ? (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: ?�1 = 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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