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2.3.1 StochasticMatrices There are several properties that can be known from a stochastic matrix. The spectral radius of a matrix is defined as: ρ(A) = max |λ|, (II.13) λεσ(A) Where σ(A) is the set of distinct eigenvalues of the matrix. The infinite norm of a matrix is defined as: ∑ |aij|, The norm creates an upper bound on ρ(A), so it is also true that: ρ(A) ≤ ||A|| ||A||∞ =max i j (II.14) (II.15) All stochastic matrices have a row sum of 1, therefore ||A||∞ = 1. Equivalently, Ae = e, where e is a vector of all ones. So for all stochastic matrices, there is the associated eigenpair (1, e), and therefore: 1≤ρ(A)≤||A||∞ =1⇒ρ(A)=1. (II.16) We can show this to be true for the matrix for the model network given in II.6. The eigenvalues for S are: λ1 = 1, λ2 = 1/3, λ3 = 6 , λ4 = 6 . (II.17) The dominant eigenvalue, where |λi| > |λj| for all j, is λ1 = 1. The eigenvectors for S are: −√ 7 − 1 √ 7 − 1 8PDF Image | MATHEMATICS BEHIND GOOGLE PAGERANK ALGORITHM
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