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14 2 ⋅ pagerank background that immediately arise. The prior definition, which is often given as the definition of PageRank, lacks the most distinguishing feature of PageRank: the adjustments themselves. It is the adjustments that define PageRank, and not the flow of importance. Certainly, the flow of importance is an important aspect of PageRank, but not its defining feature. Establishing this defining feature is where we begin. 2.1 matrix computation preliminaries Before getting to PageRank, we need some notation for operations involv- ing matrices and vectors. Unless otherwise noted, the following conventions hold: bold capital letter bold lower case letters lower case Greek letters subscripted capital letters subscripted lower case letters calligraphic capital letters A,G,P b,v,x α, β, γ Aij,Gij,Pij bi,vi,xi G, S for matrices, for vectors, for scalars, for matrix elements, for vector elements, and for graphs and sets, which is a variation on Householder notation used in a few recent text- books [Meyer, 2000; Trefethen and Embree, 2005]. The following symbols represent standard matrix or vector operations: AT,xT is the transpose of a matrix or vector, A+ is the pseudo-inverse of a matrix, e is the vector of all ones of appropriate length, ∥⋅∥ is the 1-norm of a matrix or vector, A⊗B for the Kronecker product between matrices or vectors, and A●B for the Hadamard, or elementwise, product between matrices or vectors. Horn and Johnson [1991] have a nice background on the Kronecker and Hadamard products. These are less well known than the standard product operations between matrices and have a few fascinating properties. Throughout this thesis, P represents a square, column stochastic matrix. Formally, column stochastic implies P ≥ 0 and eTP = eT. Taking P as ij column stochastic is contrary to the notation in probability, where P is a rowPDF Image | Instagram Cheat Sheet
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