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36 2 ⋅ pagerank background where D1 is a diagonal matrix for all the eigenvalues on the unit circle and J2 is a matrix of Jordan blocks for all the interior eigenvalues. Substituting the Jordan form into the PageRank equation produces (I − αP)x = (1 − α)v (I − αXJX−1)x = (1 − α)v I (I−α[ D1 J2 (I−αJ) z =(1−α) u =X−1 x =X−1 v ])z=(1−α)u. These equations are decoupled and give (1−α)z0 =(1−α)u0 (I−αD1)z1 =(1−α)u1 (I−αJ2)z2 =(1−α)u2. BothD1 andJ2 havenoeigenvaluesequalto1,andthenasα→1,z1 →0 andz2 →0;butz0 =u0 forallα=/1andinthelimit,then,z0 isstillu0. Y0 Suppose X = [ X0 X1 X2 ] and X−1 = [ Y1 ] are partitioned conformally with Y2 J. We have now established that lim x(α) = X0 Y0 v. (2.25) α→1 Although this technique makes it easy to see that the limit value exists, it is not insensitive to the formulation of the problem. For PseudoRank (problem 2) with σ = 1 the linear system ( I − α P ̄ ) y ( α ) = v has no limit for y(α) as α → 1 because the right-hand side is not normalized to be consistent. For this reason, and others, we prefer the core PageRank formulation (problem 1). 2.7.2 Jordan canonical form An alternate derivation of the limit vector uses the Jordan canonical form of M(α) instead. Serra-Capizzano [2005] proposed this idea and we repeat that derivation here to elucidate the Jordan form of M and the eigenvalues after the modification. In the derivation of the Jordan form, Serra-CapizzanoPDF Image | Instagram Cheat Sheet
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