18.06 Lecture Notes - Lecture 3: Awk, Olm, Ston

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The largest eigenvalue is real and positive and so is its eigenvector. The maximum eigenvalue controls the powers of a. Example: find i) ii) where d is a diagonal matrix and the column of u are the eigenvectors of a. Start by nding the eigenvalues of a and their corresponding eigenvectors. Note: markov matrices always have an eigenvalues of one, so start there. and nd the eigenvector by nding the solution to (a- i)x = 0 use the trace to nd the other eigenvalue. A = ud u so to get through this calculation quickly, start on the far right and work your way to the right so we can plug in for when n is in nity to get the steady state. We can write a = s s where s is the eigenvector matrix and is the diagonal eigenvalue matrix. When we raise a to a power, we get a = s s.

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