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lecture18

# lecture18 - Lecture 18 Eigenvalue Problems II Shang-Hua...

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Lecture 18 Eigenvalue Problems II Shang-Hua Teng

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Diagonalizing A Matrix Suppose the n by n matrix A has n linearly independent eigenvectors x 1 , x 2 ,…, x n . Eigenvector matrix S: x 1 , x 2 ,…, x n are columns of S. Then = Λ = - n AS S λ λ 1 1 Λ is the eigenvalue matrix
Matrix Power A k S -1 AS = Λ implies A = S Λ S -1 implies A 2 = S Λ S -1 S Λ S -1 = S Λ 2 S -1 implies A k = S Λ k S -1

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Random walks How long does it take to get completely lost? 0 0 0 0 0 1
Random walks Transition Matrix 1 2 3 4 5 6 = 0 0 0 0 0 1 0 2 1 4 1 0 0 2 1 3 1 0 4 1 0 0 0 3 1 2 1 0 2 1 3 1 0 0 0 4 1 0 3 1 0 0 0 4 1 2 1 0 2 1 3 1 0 0 0 3 1 0 100 P

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Matrix Powers If A is diagonalizable as A = S Λ S -1 then for any vector u , we can compute A k u
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lecture18 - Lecture 18 Eigenvalue Problems II Shang-Hua...

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