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used a row-stochastic P. To keep the results comparable (and to keep readers on their toes) we revert to Langville and Meyer’s notation for this subsection (which uses a row-stochastic S instead). At the end, we’ll double check that the limit value is the same when derived from the eigensystem (here) and the linear system (the last section). Recall that the Google matrix is G=αS+(1−α)evT =MT. (2.26) Let S = VJV−1 be the Jordan canonical form of S.8 We’ll state the Jordan form without the presence of a final scaling matrix to transform the off-diagonal elements in the Jordan blocks to unit values. (Recall that the choice of off- diagonal values in the Jordan blocks is arbitrary.) Without further ado, set 8 Serra uses X here, but we’ve re- placed it by V to avoid confusion with X in the previous section. and then R=I−e wT, 1 wTe =0, 1 wT = (1 − α)(eT − vT V)(I − αJ)−1 1 G = VR(αJ + (1 − α)e1 e1T )R−1 V−1 . (2.27) (2.28) (2.29) (2.30) Stated as such, this result is somewhat opaque. The derivation is straightfor- ward, but needs a few useful facts about stochastic matrices and eigenvalues. So let’s work through it. From S = VJV−1, we have V−1GV = αJ + (1 − α)V−1evT V. (2.31) We simplify the above expression through V−1e = e , which follows from the 1 fact that S is a stochastic matrix.9 At this point, we simply guess the structure of the matrix that reduces the right-hand side of the previous expression to a Jordanmatrix.LetR=I+e wT.We’llshowhowtopickwsothat 1 R−1V−1GV=(αJ+(1−α)e eT)R−1. (2.32) 11 9 To be precise, we need the prop- erty that 1 is a non-defective eigen- value of a stochastic matrix and thus the Jordan block has no off- diagonal elements. 2.7 ⋅ the limit case 37PDF Image | Instagram Cheat Sheet
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